
Oass_U 



Book 



COPYRIGHT DEPOSIT 



HAND BOOK 



OF 



CALCULATIONS 



FOR 



ENGINEERS. 




PRESS OF 

IylVINGSTON MlDDLEDITCH & CO.. 

26 CORTLANDT ST., N. Y. 



THIS WORK IS DEDICATED TO C. A. H. WITH 
FILIAL AFFECTION. 



NEW YORK: 

Copyright by Theo. Audei. & Co., 

All rights reserved. 

1890. 



HAND BOOK 

CALCULATIONS 

FOR 

ENGI NEBRS 

AND 

FIREMEN. 

RELATING TO 

THE STEAM ENGINE, THE STEAM BOILER, 
PUMPS, SHAFTING, ETC. 




Comprising the Elements of Mechanical Philosophy, Mensuration, 

Geometry, Algebra, Arithmetical Signs, and Tables. 

United States Weights, Measures and Money ; Tables of Wages, 

with Copious Notes, Explanations and Help Rules 

Useful for an Engineer. 

And for Reference, Tables of Squares and Cubes, Square and 

Cubk Roots, Circumference and Areas of Circles, Tables 

<>f Weights of Metals and Pipes, Tables of 

Pressures of Steam, Etc., Etc., Etc. 

By J^HAWKINS, M. E., 

Honorary Member National Association of Stationary Engineers ; 

Editorial Writer, Author Maxims and Instructions 

for the Boiler Room. 



THEODORE AUDEL & CO. 

Publishers, 

91 LIBERTY STREET, 

New York City. 



. \ 



V 




*>0j/. 



V 



**>' 



<* Y \ 









INTRODUCTION. 



" I would give a thousand dollars if I knew the principles 
upon which niy engine works/' 

This was the remark of a western engineer made to a gentle- 
man who was admiring the performance of the steam plant 
under the charge of the former. " I can attend to every nec- 
essary thing about my whole apparatus; engine, boilers, pumps, 
pipes, and do all that is expected of an engineer, but 1 don't 
know why the steam does its work, and I would give a thousand 
dollars to know." 

This work is prepared for those who, like the engineer whose 
words are quoted, wish to know, and are willing to pay the cost, 
in money and study. 

Abraham Lincoln once said, in the early days of his opening 
manhood, with the warm enthusiasm characteristic of his noble 
mind, " That man who furnishes me with a good book is my 
best friend;" at the age of 18 he was the proud owner of six 
volumes. 

The desire has been strong indeed, in the mind of the author, 
while compiling this work, upon a single page of which, at 
times, several days have been spent, that it might come to 
many aspiring men, with the same potent good, as the few 
books which Abraham Lincoln had access to in his early strug- 
gling days. 



viii INTRODUCTION. 



In the wide expanse of mathematics it has been a task of the 
utmost difficulty for the author to lay out a road that would 
not too soon weary or discourage the student; if he had his 
wish he would gladly advance step by step with his pupil, 
and much better explain, by word and gesture and emphasis, 
the great principles which underlie the operations of mechanics; 
to do this would be impossible, so he writes his admonition in 
two short words: In case of obstacles, " go cw." If some rule 
or process seems too hard to learn, go around the difficulty, 
always advancing, and, in time, return and conquer. 

One thing of importance may here be said. The value of a 
teacher or instructor cannot be overestimated. Men were not 
made to do their work alone; they are created so that they need 
assistance and encouragement in every direction except down- 
wards; to be helped and to help is the universal law. In no 
profession more than steam engineering does this law hold 
truer, and while the editor has written down the hard problems 
he has all the time, while making them as plain as possible to do 
so, had the secret wish that the learner might have at his side, 
when the book came to him, a kind and generous tutor, who 
could, and cheerfully would, go with him over the untravelled 
road. 

There is a single unique Book in the world, two thousand 
years compiling, of which it is said that no person can be called 
foolish who diligently peruses its pages; so the author's top- 
most wish has been now to prepare a book so elementary aud yet 
so wide in its scope that no engineer or fireman could justly be 
called ignorant who had carefully studied and become familiar 
with its pages. We quote for a motto— 

"Education does not consist merely in storing the head with 
materials; that makes a lumber room of it; but in learning how 
to turn those materials into useful products; that makes a fac- 
tory of it; and no man is educated unless his brain is a factory, 
with storeroom, machinery and material complete." 

Hence in arranging the materials of this work, the author 
has aimed to give it a certain completeness and harmony with 
itself, from beginning to end; to make it " a factory, with 



INTRODUCTION. ix 



storeroom, machinery and materials" so abundant in quantity 
and variety of stores, that it will answer every reasonable require- 
ment. 

In excuse for the extreme simplicity of some of the problems 
presented, the author owns that this feature of the book was 
incorporated in it through its having come to his knowledge, 
through a member of the board of U. S. government exami- 
ners, some years since, that some score of high class marine 
engineers had come very near losing their positions on account 
of their ignorance of many of the simplest items of information 
relating to their duties. 

It was in consequence of an order received from headquarters 
at Washington for the re-examination of all the engineers in a 
certain department, in the Eastern division of bhe marine ser- 
vice; the order was peremptory, and the examinations to the 
number of 60 or 70 were held forthwith. 

And it was a disagreeable fact that while few, or none, were 
really displaced, the positions of all these really competent engi- 
neers were in danger of being forfeited, because they had for- 
gotten the little things they had acquired in earlier days. 

Hence the truly wise student of this hand-book, even if of 
•established reputation, will not despise the elementary rules 
and examples presented. Nor must the humble beginner de- 
spair of the most difficult. Both extremes will be found in 
the completed volume. 

N. H. 

New York City, June 1st, 1889. 



PLAN OF THE WORK. 



The leading idea intended to be illustrated in the following- 
successive "parts " or chapters is this: that in an informal and 
not too "dry"' a method, engineers or those aspiring to be 
such shall be taught to figure the problems relating to the steam 
engine and boiler; the steam pump; shafting and pulleys; and 
all other calculations required in the varied duties of steam- 
engineering in its most intelligent and useful practice. 

The first four or five parts of the work will be occupied ex- 
clusively with what may be called the general principles ofmath- 
e??m^'cs— principles which are used in all times and places and 
in an infinite variety of machines, and their application to the 
use of man. Next, these elements will be illustrated by the 
practice of to-day in steam engineering in its various depart- 
ments. Eules for calculating horse power of engines and 
boilers will be given in the plainest manner and fully illustra- 
ted by diagrams; rules for figuring the safety-valve pressure of 
boilers, strength of materials, size and capacity of pumps, etc., 
etc., with help rules, notes and remarks based upon the most 
approved practical experience. 

The work will close with valuable and copious tables of roots 
and powers of numbers, and diameters and circumferences of cir- 
cles, and all the data commonly found in the most advanced 
works written for mechanics; hence, the first part of the work, 
perhaps three quarters of it, will be for instruction and the 
other part for reference. 

It must not be forgotten that the elements only of arithmetic, 
geometry, algebra, mensuration, etc., are to be introduced in 
the work, but it is upon these elements that the whole structure 
of mathematics rests, and form the groundwork where the most 
advanced and the most lowly beginner can meet with mutual 
respect. 



PLAN OF THE WORK. XI 

It is planned that the ultimate result of this publication will 
be the compiling of a standard and valuable volume, contain- 
ing all the mathematics relating to steam engineering necessary 
for an intelligent engineer in his daily practice; hence the 
author, ere the work proceeds too far, will be pleased to receive 
the helpful suggestions of his kindly reader as to the most 
desirable contents for such a comprehensive work. 

For the space it occupies the explanation of the use of form- 
tilas or forms will be found to be most useful to the practical 
man, as it teaches him the school language of expressing calcu- 
lations. This custom is the same as that followed by the 
physician in writing aqua pura instead of "pure water"; and 
the gardener giving Latin names to his plants instead of plain 
English terms. The use of formulae is so universal that many 
publications, otherwise of great value to the engineer, are to 
him as a sealed book; but with the explanations to be found in 
this work a great part of the difficulty will be obviated. 

At the issue of Part 1 the whole work is in manuscript, but 
it will be printed in 10 monthly parts. This is to accommodate 
the student, to whom a single Paet will be the moderate allow- 
ance for a month's study, and also to allow snch changes as may 
seem necessary to perfect the plan of the work before it is 
advanced to book form. 

The index of the whole book will be issued with the last 
number, in convenient shape, and at that time a more formal 
preface will be written, in which due acknowledgement will be 
made for assistance from persons and authors whose advice and 
experience has been drawn upon. 

One other item may be added, but not enlarged upon: that 
is the desire to give for a moderate cost, information of large 
value to the purchaser. An engineer who can figure and do it 
correctly is of more value than one who cannot, and this esteem 
is (between the reader and the author) expressed by larger com- 
pensation and longer service in one position. 



Hand Book of Calculations. 



ARITHMETICAL SIGNS. 



The principal characters or marks used in arithmetical com- 
putations to denote some of the operations, are as follows : 

= Equal to. The sign of equality; as 100 cti. = $1— signi- 
fies that one hundred cents are equal to one dollar. 

— Minus or Less. The sign of subtraction; as 8-2=6, that 
is, 8, less 2, is equal to 6. 

+ Plus or More. The sign of addition; as 6+8=14; that 
is, 6 added to 8, is equal to 14. 

X Multijiliedby. The sign of multiplication; as 7x7=49; 
that is, 7 multiplied by 7 is equal to 49. 

-^ Divided by. The sign of division; as 16-^4 4; that is, 
16 divided by 4 is equal to 4. 

There are still other characters and marks which will be 
added as needed as the work progresses, but these are the prin- 
cipal ones. 



ARITHMETICAL FORMULAS. 



An arithmetical formula is a general rule of arithmetic 
expressed by signs. 

The following 10 formulas include the elementary operations 
of arithmetic and follow from the succeeding illustrations. 

1. The Sum=«£? the parts added. 

2. The Differences/^ Minuend — the Subtrahend. 

3. The Minuend = the Subtrahend -f- the Difference. 

4. The Subtrahend = the Minuend — the Difference. 

5. The Product = the Multiplicand x the Multiplier. 

6. The Multiplicand = the Product + the Multiplier. 

7. The Multiplier = the Product -h the Multiplicand. 

8. The Quotient = the Dividend -h the Divisor. 

9. The Dividend = the Quotient x the Divisor. 
10. The Divisor = the Dividend -^ the Quotient. 
Formulas or formulas, express the plural of formula — a Latin 

word which means, simply, a form; hence a formula is a form 
of stating a problem . 



Hand Book of Calculations. 



ARITHMETIC. 



Arithmetic is the science or orderly arrangement of numbers 
and their application to the purposes of life. The processes of 
arithmetic are merely expedients for making easier the discov- 
ery of results, which every mechanic of ordinary ingenuity 
would find a means for discovering himself, if really called upon 
to set about the task, for it is possible for a man to be a good 
working engineer, and at the same time be quite ignorant of 
reading, writing or figuring; but experience shows that in order 
to advance in the confidence of others, it is very necessary to 
know something of the elements, or first things, of mathemat- 
ics related directly or indirectly to steam. 

Arithmetic is the science of numbers, and numbers treat of 
magnitude or quantity. Whatever is capable of increase or 
diminution is a magnitude or quantity; a sum of money, a 
weight, or a surface, is a quantity, being capable of increase or 
diminution. But as we cannot measure or determine any quan- 
tity, except by considering some other quantity of the same 
kind as known, and pointing out their mutual relation, the 
measurement of quantity or magnitude is reduced to this: 

Fix at pleasure upon any known kind of magnitude of the 
same species as that which has to be determined, and consider it 
as the measure or unit. 

If, for example, we wish to determine the magnitude of a 
sum of money we must take some piece of known value, as a 
dollar, which is the unit of money, and show how many such 
pieces are contained in the given sum. 



j 4 Hand Book of Calculations. 

The foot rule is the unit or measure of length most used for 
engineering purposes; the foot is divided into twelve inches and 
the inch is subdivided in half inches, quarter inches, eighths 
and sixteenths It is plain that into whatever number of parts 
the inch is divided, wc shall equally have the whole inch if we 
take the whole of the parts of it; if it were divided into ten 
equal parts, then ten of these parts would make an inch. 

The unit of surface in steam engineering is represented by 
the square inch. 

The unit of time is in usual practice one minute; thus we say 
an engine makes so many revolutions per minute, and its per- 
formance is based upon that. 

The unit of work is the force required to raise one pound, 
one foot high from the earth, in the atmosphere, no time being 
taken in the account; it is known as the foot pound. 

Atmospheric pressure at the sea level is the unit of pressure. 

The unit of heat is the amount of heat required to raise one 
pound of water one degree, usually from 32° to 33° Fahr. 

The unit of numbers is the figure one (1). 

These references to the different measures, or units, are made 
in view of their frequent use in ascertaining duties performed 
by steam engines and boilers. They enter into all engineering 
calculations in connection with Tables to be found elsewhere 
in this volume, and their utility will be clearly explained and 
readily understood from their combination with practical cal- 
culations elsewhere found in this volume. 

Electric units. The unit of electric force is the volt ; 
the unit of resistance is the ohm; the unit of current strength 
or volume, is the ampere; the unit of current quantity consid- 
ered with reference to time is the coulomb; the electric unit of 
capacity is the farad; the unit of electric power is the watt, etc. 

The measurement of electricity is one of the newest discover- 
ies, to which a separate space will herewith be devoted, in which 
the electric units of force, resistance, etc., will enter into the 
practical problems relating to electric lighting. 



Hand Book of Calculations. 15 



NOTATION AND NUMERATION. 



Notation in Arithmetic is the writing down of figures to ex- 
press a number or numbers, and Numeration is the reading of 
numbers already written. 

There are nine figures — 1, 2, 3, 4, 5, 6, 7, 8 and 9 nsed in 
arithmetic, and the (nanght) to represent nothing. 

The nnmber 1 is called the unit. The number 9 is a collec- 
tion of nine of these nnits. 

By means of these 10 figures we can represent any number. 
When one of the figures stands by itself, it is called a unit; but 
if two of them stand together, the right hand one is still called 
a unit, but the left hand one is called tens; thus, 79 is a collec- 
tion of 9 units and 7 sets of ten units each, or of 9 units and 70 
units, or of 79 units, and is read as seventy-nine. 

If three of them stand together, then the left hand one is 
called hundreds; thus 279 is read two hundred and seventy- 
nine. 

To express larger numbers other orders of units are formed, 
the figure in the 4th place denoting thousands; in the 5th place 
ten thousands; these are called units of the fifth order. 

The sixth place denotes hundred thousands, the seventh place 
denotes millions, etc. 

The French method (which is the same as that used in the 
IT. S.) of writing and reading large numbers is shown in the 



following 



Names of 
periods. 



NUMERATION TABLE. 
Billions. Millions. Thousands. Units. Thousandths. 



CO 

Go* 3 

Order of B » % g S B . I si 

Units. 5§ J J I I -3 « & 5 1 

t3.S. ^-B« p d3S ^3 ,_, ^3 

^.og 'OdO '■dtSg ns . aj g ,B^2 



rHi.rt fl ' 1H c- 11 ^ etc-*-* .,H +j ,-1 

3 ? ^ § B :3 § fl 2 3 B a « fl ^ - 

S H PQ PhH^ Mhh MHh^ ft HWh 

87 6, 54 3, 20 1, 28 2, . 489 



16 Hand Book of Calculations. 

The number in the table is read eight hundred and seventy 
six billion, five hundred and forty-three million, two hundred 
and one thousand, two hundred and eighty-two, and four hun- 
dred and eighty-nine thousandths. 

To express larger numbers other periods are formed in like 
manner, called Trillions, Quadrillions, Quintiliions, Sextillions, 
Septillions, Octillions, Nonillions, Decillions. Each of these 
periods increase the values of all the figures to which it is added 
1,000 times. 

Figures are always read from left to right; thus, one million, 
one thousand and two is in figures 1,001,002, the figures 1, 1, 
and 2 occupying the 7th, the 4th and the 1st place, and cyphers 
the intermediate spaces. The ' ' one million ".at the left.is read 
first and the unit " two " at the right is read last— and this is 
the universal rule with the important exception of decimals, 
hereafter explained. 

In the table given it will be observed that the long row of 
figures are divided by commas (,). This is to aid in their ready 
reading. The first set is called units, the second thousands, 
the third millions, etc. 

Beginning at units place, the orders on the right of the deci- 
mal point, express tenths, hundredths, thousandths, etc. 

Examples for Peactice. 

Notation. Write in figures eight million, two hundred fifty- 
nine thousand eight hundred and ninety-two. 
Ans. 8,259,892. 

2. Write four hundred and sixty-two thousand and nine. 
Ans. 462,009. 

3. Write four billion, four million, four thousand and four. 
Ans. 4,004,004,004. 

4. Write six hundred and two. 

5. Write sixteen thousand, seven hundred and ninety two. 

6. Write six hundred and eight thousand four hundred and 
seventy-nine. 



Hand Book of Calculations. ij 

Numeration. Read the following numbers: 

1. 19. 

2. 406. 

3. 9,206. 

4. 90,009. 

5. 896,724. 

6. 7,428,940. 

7. 63,178,392. 

This system is called Arabic Notation from the fact that it 
was introduced into Europe in the 10th century by the Arabs. 

Its great law is that ten units in any order make one unit of 
the next order. 

And the moving a figure one place either increases or dimin- 
ishes its value by the uniform scale of ten. 

Hence it is called the Decimal system from the Latin word 
decern, which means ten. 



ROMAN NOTATION. 

This is the method of expressing numbers by letters. 
I, V, X, L, O, D, M, 

1, 5, 10, 50, 100, 500, 1,000 

1. Repeating a letter repeats its value, thus: 1=1, 11=2. 

2. Placing a letter of less value before one of greater valu^ 
diminishes the value of the greater by the less; thus, IV=4, 
IX=9, XL=40. 

3. Placing the less after the greater increases the value of 
the greater by that of the less; thus, YI=6, XI=11, LX=60. 

4. Placing a horizontal line over a letter increases its value 
a thousand times; thus, 1V=4000, M=l, 000,000. 



i8 



Hand Book of Calculations. 



ADDITION TABLE. 



1 

1 and 




2 and 




3 and 




4 and 




5 and 


1 


are 2 


1 


are 3 


1 


are 4 


1 


are 5 


1 


are 6 


2 


" 3 


2 


" 4 


2 


" 5 


2 


" 6 


2 


a >y 


3 


" 4 


3 


" 5 


3 


" 6 


3 


cc 7 


3 


" 8 


4 


" 5 


4 


" 6 


4 


a w 


4 


" 8 


4 


" 9 


5 


" 6 


5 


ftf w 


5 


" 8 


5 


" 9 


5 


" 10 


6 


a >y 


6 


" 8 


6 


" 9 


6 


" 10 


6 


" 11 


7 


" 8 


7 


" 9 


7 


" 10 


7 


" 11 


7 


" 12 


8 


" 9 


8 


« 10 


8 


" 11 


8 


" 12 


8 


" 13 


9 


" 10 


9 


" 11 


9 


" 12 


9 


" 13 


9 


« !4 


10 


" 11 


10 


" 12 


10 


" 13 


10 


" 14 


10 


" 15 


6 and 




7 and 




8 and 




9 and 


10 and 


1 


are 7 


1 


" 8 


1 


are 9 


1 


are 10 


1 


are 11 


o 


" 8 


2 


" 9 


2 


" 10 


2 


" 11 


2 


" 12 


3 


" 9 


3 


" 10 


3 


" 11 


3 


" 12 


3 


" 13 


4 


" 10 


4 


" 11 


4 


" 12 


4 


" 13 


4 


« 14 


5 


« n 


5 


" 12 


5 


" 13 


5 


« 14 


5 


" 15 


6 


" 12 


6 


" 13 


6 


" 14 


6 


" 15 


6 


" 16 


7 


" 13 


7 


" 14 


7 


" 15 


7 


" 16 


7 


« 17 


8 


« 14 


8 


" 15 


8 


" 16 


8 


" 17 


8 


" 18 


9 


" 15 


9 


" 16 


9 


« 17 


9 


" 18 


9 


" 19 


10 


i( 16 


10 


« 17 


10 


" 18 


10 


" 19 


10 


" 20 



Hand Book of Calculations. ig 



ADDITION. 



The first process of arithmetic is Addition; and here the first 
steps are made by counting upon the fingers as an aid to the 
perceptions of the total amount of the quantity that has to be 
-expressed. Persons even of considerable mathemetical experi- 
ence will often find themselves counting their fingers, or press- 
ing them down successively on the table in order to assist their 
memory in performing addition. 

For example, if we hold up 5 fingers of one hand and 3 of 
the other and are asked how much 5 and 3 amount to we at 
once see that the number is 8, as we actually or mentally count 
the other three fingers from 5. 

But the best course is to commit very thoroughly to memory 
an addition table, just as the multiplication table is now com- 
monly committed to memory by arithmetical students. A ta- 
ble of this kind is here introduced, and it should be gone over 
and over again until its indications are as familiar to the mem- 
ory as the letters of the alphabet, and until the operation of 
addition can be performed without the necessity of mental effort. 
The table is so plain as scarcely to require explanation. The 
sign of addition is -f- It is called plus, or more. 

The sum or amount of any calculation no matter how small 
or large contains as many units as all the numbers added. 

Addition is uniting two or more numbers into one. The 
result of the addition is called the Sam or Amount. In addi- 
tion the only thing to be careful about except the correct doing 
of the sum, is to place the unit figures under the unit figure 
above it, the tens under the tens, etc. 

Kule. 

After writing the figures down so that units are under units, 
tens under tens, etc. : 

Begin at the right hand, up and down row, add the column 
.and write the sum underneath if less than ten. 



20 Hand Book of Calculations. 

If however the sum is ten or more write the right hand figure 
underneath, and add the number expressed by the other figure 
or figures with the numbers of the next column. 

Write the whole of the last column. 



Examples for Practice. 

7,060 248,124 13,579,802 

9,420 4,321 83 

1,743 889,876 478,652 

4,004 457,902 87,547,289 



22,227 Ans. 

Use great care in placing the numbers in vertical lines, as 
irregularity in writing them down is the cause of mistakes. 

Rule eor Proving the Correctness of the Sums, 

Add the columns from the top doivnward, and if the sum is 
the same as when added up then the answer is right. 

Add and prove the following numbers : 
684 32 257 20. Ans. 993. 
42 89 22 99 ? 
1006 7008 01 62 ? 

TABLE OF UMTS. 

The unit of money in the U. S. is one dollar. 
The unit of length is one foot. 

The unit of surface is the square inch. 
The unit of toork is the foot pound. 

The unit of heat is one degree, Fahrenheit. 
The unit of numbers is the figure 1. 
The unit of electric power is the watt. 



Hand Book of Calctilations. 



21 



SUBTRACTION TABLE. 



1 from 


2 from 


3 from 


4 from 


5 from 


1 leaves 


2 leaves 


3 leaves 


4 leaves 


5 leaves 


2 " 1 


3 < 


' 1 


4 " 1 


5 " 1 


6 " 1 


3 " 2 


4 • 


' 2 


5 " 2 


6 " 2 


7 " 2 


4 " 3 


5 ' 


' 3 


6 " 3 


7 " 3 


8 " 3 


5 " 4 


6 ' 


' 4 


7 " 4 


8 « 4 


9 " 4 


6 " 5 


7 ' 


< 5 


8 " 5 


9 " 5 


10 " 5 


7 " 6 


8 ' 


4 6 


9 " 6 


10 " 6 


11 " 6 


8 " 7 


9 ■ 


' 7 


10 " 7 


11 " 7 


12 « 7 


9 " 8 


10 < 


4 8 


11 " 8 


12 " 8 


13 " 8 


10 " 9 


11 » 


• 9 


12 " 9 


13 " 9 


14 " 9 


11 " 10 


12 ' 


' 10 


13 " 10 


14 " 10 


15 " 10 


6 from 


7 from 


8 from 


9 from 


10 from 


6 leaves 


7 leaves 


8 leaves 


9 leaves 


10 leaves 


7 " 1 


8 < 


• 1 


9 « 1 


10 


< 1 


11 " 1 


8 " 2 


9 ' 


' * 


10 " 2 


11 


■< 2 


12 "2 


9 " 3 


10 ' 


< 3 


11 " 3 


12 


'< 3 


13 " 3 


10 "4 


11 


' 4 


12 " 4 


13 


'< 4 


14 " 4 


1 11 " 5 


12 ' 




13 " 5 


14 


' 5 


15 « 5 


12 " 6 


13 ' 


' 6 


14 " 6 


15 


< 6 


16 " 6 


13 " 7 


14 ' 


■ 7 


15 " 7 


16 


' 7 


17 " 7 


14 « 8 


15 ' 


' 8 


16 " 8 


17 


< 8 


18 " 8 


15 " 9 


1G < 


' 9 


17 " 9 


18 


< 9 


19 " 9 


16 " 10 


17 ' 


' 10 


18 " 10 


19 ' 


< 10 


20 " 10 



22 Hand Book of Calculations. 



SUBTRACTION. 



Subtraction is taking one number from another. 

As in addition, care must be used in placing the units under 
the units, the tens under the tens, etc. 

The answer is called the remainder or the difference. 

The sign of subtraction is ( — ) Example: 98 — 22=76. 

Subtraction is the opposite of addition — one " takes from 9> 
while the other "adds to." 



Rule. • 

Write down the sum so that the units stand under the units,, 
the tens under the tens, etc., etc. 

Begin with the units, and take the under from the upper fig- 
ure and put the remainder beneath the line. 

But if the lower figure is the largest add ten to the upper 
figure, and then subtract and put the remainder down — this bor- 
rowed 10 must be deducted from the next column of figures, 
where it is represented by 1. 



Examples eoe Pkactice. 

892 89,672 89,642,706 

46 46,379 48,765,421 



846 remainder. 



Note. 



In the first example 892—46 the 6 is larger than 2; borrow 10,. 
which makes it twelve, and then deduct the 6; the answer is 6. 
The borrowed 10 reduces the 9 to 8, so the next deduction is 4 
from 8=4 is the answer. 



Hand Book of Calcidations. 2j 



Kule for Proving the Correctness of the Subtraction. 

Add the remainder, or difference, to the smaller amount of 
the two sums and if the two are equal to the larger, then the 
subtraction has been correctly done. 
Example. 898 246 

246 Now then, 652 

652 898 correct Ans. 



Examples, Consisting of Notation, Addition and Sub- 
traction. 

1. Add together twenty-seven thousand four hundred and 
twenty-eight; ninety-one thousand eight hundred and seventy- 
nine; sixty-five thousand two hundred and fifty-nine ; and 
thirty-seven thousand and eight. Ans. 221,574. 

2. Add seven hundred billions, nine hundred and one thou- 
sand; forty millions thirty thousand and ten; five hundred 
thousand; eight hundred and ninety-one millions and twelve; 
twenty-four millions two hundred and one thousand and six 
hundred and forty-four ; and two hundred and ninety-three 
billions, nine hundred and ninety-two millions, eight hundred 
and sixty-seven thousand, three hundred and twenty-nine; 
five billions, fifty millions, five' hundred thousand and five. 
Ans. 1,000 Billions, or 1 Trillion. 

Note. — This sum is best done by the aid of the numeration 
table. It is given for practice to form a habit of accuracy in 
doing long calculations. 

3. From sixty-four thousands two hundred and ten millions 
nine hundred and twenty thousands six hundred and fifty-one: 
take twenty-nine thousand five hundred and fifty-four mil- 
lions, three hundred and seventy-four thousand six hundred 
and eighty-eight. Ans. 34,656,545,963. 

4. From ninety billions, four hundred millions, seven thou- 
sand and six: take nine billions, one hundred millions, five 
thousand nine hundred and fifty-six. Ans. 81,300,001,050. 



24 



Hand Book of Calculations. 



MULTIPLICATION TABLE. 



Once 


1 is 1 


2 " 2 


3 " 3 


4 " 4 


5 " 5 


6 " 6 


IV « IV 


8 " 8 


9 " 9 


10 " 10 


11 "11 


12 " 12 



2 times 

1 are 2 

2 " 4 

" 6 
" 8 
"10 
"12 
"14 
"16 
"18 
"20 
"22 
"24 



3 times 

1 are 3 

2 " 6 
" 9 
" 12 
"15 
"18 
"21 
"24 
"27 
"30 
"33 
"36 



3 

4 

5 

6 

7 

8 

9 

10 

11 

12 



4 times 


1 


are 4 


2 


" 8 


3 


"12 


4 


" 16 


5 


"20 


6 


"24 


7 


"28 


8 


"32 


9 


"36 


10 


"40 


11 


"44 


12 


"48 



5 times 

1 are 5 

2 " 10 

" 15 
"20 
"25 
"30 
"35 

8 "40 

9 "45 
50 
55 
60 



7 times 

1 are 7 

2 " 14 

3 "21 

4 "28 

5 "35 

6 "42 

7 "49 

8 "56 

9 "63 

10 " 70 

11 " 77 

12 "84 



8 times 


1 


are 8 


2 


"16 


3 


"24 


4 


"32 


5 


"40 


6 


"48 


7 


"56 


8 


"64 


9 


ee iyo 


10 


"80 


11 


"88 


12 


" 96 



9 times 


1 


are 9 


2 


a 


18 


3 


a 


27 


4 


a 


36 


5 


ee 


45 


6 


ee 


54 


7 


ee 


63 


8 


ee 


72 


9 


ee 


81 


10 


(C 


90 


11 


ee 


99 


12 


"108 



10 times 


1 


are 10 


2 


i i 


20 


3 


ee 


30 


4 


ee 


40 


5 


ce 


50 


6 


i( 


60 


7 


a 


70 


8 


a 


80 


9 


ee 


90 


10 


a 


100 


11 


ee 


110 


VI 


ee 


120 



11 times 
1 are 11 



2 " 

3 " 

4 " 

5 " 

6 " 

7 " 

8 " 

9 " 

10 "110 

11 "121 

12 "132 



22 
33 
44 
55 

66 

77 
88 
99 



6 times 

1 are 6 

2 "12 

3 "18 

4 "24 

5 "30 

6 "36 

7 "42 

8 "48 

9 "54 

10 " 60 

11 " 66 

12 " 72 



12 times 

1 are 12 

2 " 24 

3 " 36 

4 " 48 

5 " 60 

6 " 72 

7 " 84 

8 " 96 

9 "108 

10 "120 

11 "132 

12 "144 



Hand Book of Calczdations. 25 



MULTIPLICATION. 



Multiplication is finding the amount of one number increased 
as many times as there are units in another. 

The number to be multiplied or increased is called the Mul- 
tiplicand. 

The Multiplier is the number by which we multiply,, It 
shows how many times the multiplicand is to be increased. 

The answer is called the Product. 

The multiplier and multiplicand which produce the product 
are called its Factors. This is a word frequently used in math- 
ematical works and its meaning should be remembered. 

The sign of multiplication is X and is read i( times " or mul- 
tiplied by; thus 6 X 8 is read, 6 times 8 is 48, or, 6 multiplied 
by 8 is 48. 

The principle of multiplication is the same as addition, thus 
3x8=24 is the same as 8+8+8=24. 

Eule fob Multiplying. 

Place the unit figure of the multiplier under the unit figure 
of the multiplicand and proceed as in the following: 

Examples. Multiply 846 by 8; and 487,692 by 143. 
Arrange them thus: 

487,692 
143 

846 ■ 

8 1463076 

1950768 

6,768 487692 



69,739,956 



26 Hand Book of Calculations. 



But if the multiplier has ciphers at its end then place it a& 
in the following: 

Multiply 83567 by 50; and 898 by 2800. 

898 
2800 



83567 718400 

50 1796 



4,178,350 2,514,400 

Examples foe Practice. 

1. Multiply 4,896,780 by 9. 

2. " 94,200,642 " 12. 

3. " 843,217,896 " 800. 

4. " 4,980 " 1,276. 

5. " 76 " 7,854. 

6. « 34 5 571,248 " 9,876. 

The product and the multiplicand must be in like numbers. 
Thus, 10 times 8 gallons of oil must be 80 gallons of oil. 4 
times 5 dollars must be 20 dollars; hence the multiplier must 
be the "number and not the thing to be multiplied. 

In finding the cost of 6 tons of coal at 7 dollars per ton the 7 
dollars are taken 6 times, and not multiplied by 6 tons. 

When the multiplier is 10, 100, 1000, etc., the product may 
be obtained at once by annexing to the multiplicand as many 
ciphers as there are in the multiplier. 

Example. 

1. Multiply 486 by 100. 

Now 486 with 00 added=48,600. 

2. 6,842 X 10,000=how many ? Ans. 68,420,000. 

To prove the result in multiplication multiply the multiplier 
by the multiplicand, and if the product is the same in both cases 
then the answer is right. 



Hand Book of Calculations. 



27 



DIVISION TABLE. 



1 in 




2 in 




3 in 




4 in 


5 in 


1 , 1 time 


2, 


1 time 


3, 


1 time 


4. 


1 time 


5, 1 time 


2, 2 times 


4, 


2 times 


6, 


2 times 


8, 


2 times 


10, 2 times 


3, 3 " 


6, 


3 " 


0, 


3 « 


12, 


3 " 


15, 3 " 


4, 4 " 


8, 


4 « 


12, 


4 " 


16, 


4 '< 


20, 4 " 


5, 5 " 


10, 


5 " 


15, 


5 " 


20, 


5 " 


25, 5 " 


6, 6 " 


12, 


6 " 


18, 


6 " 


24, 


6 " 


30, 6 " 


7,7 " 


14, 


7 « 


21, 


7 " 


28, 


7 " 


35, 7 " 


8, 8 " 


16, 


8 " 


24, 


8 " 


32, 


8 '< 


40, 8 " 


9, 9 " 


18, 


9 " 


2?, 


9 " 


36, 


9 " 


45, 9 (i 


10,10 " 


20, 


10 " 


30, 


10 " 


40, 


10 " 


50,10 " 


6 in 




7 in 




8 in 




d in 


10 in 


6, 1 time 


7, 


1 time 


8, 


1 time 


0, 


1 time 


10, 1 time . 


12, 2 times 


14, 


2 times 


16, 


2 times 


18, 


2 times 


20, 2 times 


18, 3 " 


21, 


3 " 


24, 


3 " 


it, 


3 " 


30, 3 " 


24, 4 " 


28, 


4 " 


32, 


4 " 


36, 


4 " 


40, 4 " 


30, 5 " 


35, 


5 " 


40, 


5 " 


45, 


5 " 


50, 5 " ' 


36, 6 " 


42, 


6 " 


48, 


6 " 


54, 


6 " 


60, 6 " 


42, 7 " 


49, 


7 " 


56, 


7 " 


63, 


7 " 


70, 7 " 


48, 8 « 


56, 


8 " 


64, 


8 " 


Tl, 


8 " 


80, 8 " : 


54, 9 " 


63, 


9 " 


72, 


9 '< 


81, 


9 " 


90, 9 " 


GO, 10 " 


70, 


10 " 


80, 


10 " 


00, 


10 " 


100,10 " 



28 Hand Book of Calculations. 



DIVISION. 



When one number has to be divided by another number the 
first one is called the dividend, and the second one the divisor, 
and the result or answer is called the quotient. 

1. To divide any number up to 12. Put the dividend down 
with the divisor to the left of it, with a small curved line sep- 
arating it, as in the following: 

Divide by 6)7,865,432 



1,310,905—2 

Here at the last we have to say 6 into 32 goes 5 times and 2 
over; always place the number that is over as above, separated 
from the quotient by a small line or else pat it as a fraction, 
thus 2/6, the top figure being the remainder and the bottom 
figure the divisor, when it should be put close to the quotient; 
thus— 1,310, 905f. 

2. To divide by any number up to 12 with a cipher or ciph- 
ers after it as 20, 70, 90, 500, 7,000, etc. 

Place the sum down as in the last example, then mark off 
from the right of the dividend as many figures as there are 
ciphers in the divisor; also mark off the cyphers in the divisor; 
then divide the remaining figures by the number remaining in 
the divisor; thus: — • 

Example. 

Divide 9,876,804 by 40. 

40)9,876,804 



246,920—4 

The 4 cut off from the dividend is put down as a remainder, 
or it might have been put down as -h or T V. 



Hand Book of Calculations. 29 



Example. 



Divide 129,876,347 by 1200. 

1200)129,876,347 



108,230—347 or Aft. 
Here there is a remainder of 3 and 47 cut off. The three 
must always be put before the 47 making it a remainder of 347 
altogether. 

3. To divide by any number that can be broken up into tivo 
factors as 18, 24, 36, 72, 144, etc. 18 is 3 times 6; then 3 and 
6 arc called factors of 18; twice 9 are 18, then 2 and 9 are also 
factors of 18 Generally any two numbers which when multi- 
plied together come to the given number, are called factors of 
that given number. 

Example. 
Divide 868.224 by 24. 
Here 4 times 6=24, therefore 4 and 6 are the factors. 

Divide first by 4 and then the quotient by 6 as follows : 

4)868,224 



24 



6)217,056 



36,176 
Example. 

Divide 9,824.671 by 63. 

63=7 times 9. ( 7)9,824,671 

>3^ 



63 1 y 10 

( 9)1,403,524—3 



155,947—1 

Here after the division by 7 there are 3 over; and after the 
division by 9 there is 1 over. What is the full remainder for 
the sum ? To find the full remainder, multiply the first divisor 
by the last remainder and add the first remainder. 

That is 7 multiplied bv 1=7. and 3 added to 7=10. 



qo Hand Book of Calculations. 



4. To divide by any number not included in the last three 
.cases. 

This is common long division as it is called. 

Eule. 

Write the divisor at the left of the dividend and proceed as 
in the following: 



Example. 



Divide 726,981 by 7,645. 
7,645)726981(95 

68805 



38931 
38225 



706 Ans. 95y\°A. 

Examples eor Exercise. 

1.— 76,298,764,833 by 9. 

•2.-120,047,629,817 " 20. 

3.— 9,876,548,210 " 48. 

4._ 3,247,617,219 " 63. 

5._ 7,140,712,614 " 41. 

6.— 329,817,298 " 107. 

7.-247,698,672,437 " 987. 

8.— 2,610,014,723 " 2406. 

9.— 10,781,493,987 " 7854. 

Multiplying the dividend, or dividing the divisor by any 
number, multiplies the quotient by the same number. 

Dividing the dividend, or multiplying the divisor by any 
number, divides the quotient by the same number. 

Dividing or multiplying both the dividend and divisor by the 
same number does not change the quotient. 



Hand Book of Calculations. ji 

TABLES OF WEIGHTS AND MEASURES 
REQUIRED BY ENGINEERS. 



AVOIRDUPOIS, or ORDINARY COMMERCIAL WEIGHT. 

This table is used for nearly all articles estimated by weight, 
except gold, silver and jewels. 

Table. 

16 drams (dr.) make 1 ounce, oz. 

16 ounces, 1 pound, lb. 

25 pounds, 1 quarter, qr. 

4 quarters or 100 lbs., 1 hundred-weight, cwfc. 

20 hundred-weight, 1 ton, T. 

LONG- MEASURE, or LINEAR MEASURE. 

This is used in estimating distances and the length of articles 

Table. 

12 inches (in.) make 1 foot ft. 

3 feet, 1 yard, yd. 

5 \ yards, 1 rod, rd. 

40 rods, 1 furlong, fur. 

8 furlongs, 1 common mil , m. 

SURFACE, or SQUARE MEASURE. 

This is used in estimating surfaces. 

Table. 

144 square inches (sq. in.) make 1 square foot, sq. ft. 

9 square feet, 1 square yard, sq. yd. 

30£ square yards, 1 square rod or perch, P. 

100 square rods or perches, 1 acre, A. 

640 acres, 1 square mile, M. 



j2 Hand Book of Calculations. 

MEASURE OF CAPACITY, or LIQUID MEASURE. 

This is used in measuring all kinds of liquids. 

Table. 

4 gills (gi. ) make 1 pint, pt. 

2 pints 1 quart, qt. 

4 quarts 1 gallon, gal. 

DRY MEASURE. 

This is used in measuring grain, roots, fruit, coal, etc. 

Table. 

2 pints (pt.) make 1 quart, qt. 

8 quarts, 1 peck, pk. 

4 pecks, 1 bushel, bu. 

Note. 
The chaldron, a measure of 36 bushels, formerly employed 
with some kinds of coal, is now seldom used. 

CIRCULAR MEASURE. 

This is used for measuring angles. 

Table. 

60 seconds (") make 1 minute, ' 

60 minutes, 1 degree, ° 

360 degrees, 1 circum., C. 

Note. 

The circumference of every circle, whatever, is supposed to 
be divided into 360 equal parts, called degrees. 

A degree is ^ of the circumference of any circle, small or 
large. 

A quadrant is a fourth of a circumference, or an arc of 90 
degrees. 

A degree is divided into 60 parts called minutes expressed by 
sign (') and each minute is divided into 60 seconds expressed by 
(") so that the circumference of any circle contains 21,600 
minutes, or 1,296,000 seconds. 



TABLE OF WAGES. 





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34 Hand Book of Calculations. 



SOLID MEASURE, or CUBIC MEASURE. 

This is used in measuring bodies, or things having length, 
breadth and height or depth. 

TABLE. 

1728 cubic inches (cu. in.) make 1 cubic foot, cu. ft. 

27 cubic feet, 1 cubic yard, cu. yd. 

128 cubic feet, 1 cord, C 



TROY WEIGHT. 

This is used for weighing gold, silver and jewels. 

Table. 

24 grains (gr. ) make 1 pennyweight, pwt. 

20 pennyweights, 1 ounce, oz. 

12 ounces, 1 pound, lb. 

A caret, for gold- weight, is 4 grains; for diamond-weight, is- 
3.2 grains. 

APOTHECARIES WEIGHT. 

SOLID MEASUEE. FLUID MEASUKE. 

20 Grains (gr)=l scruple (sc) 60 minims or drops =1 fluid dram 

3 Scruples=:l dram (dr.). 8 Fluid drams=l fluid ounce. 

8 Drams = 1 ounce (oz.). 1.6 Fluid ounces =1 pint. 

13 Ounces= 1 pound (lb.). 8 Pints=l gallon. 



Note. 

Apothecaries, in mixing medicines, usa the pound, ounce (oz.), 
and grain, of this weight; but divide the ounce into 8 drams 
(dr.), each equal to three scruples (sc), each scruple being equal 
to 20 grains. 



Hand Book of Calculations. 



35 



TIME MEASURE. 

Time is used in measuring portions of duration. 

Table. 

60 seconds (sec.) make 1 minute, 
60 minutes, 
24 hours, 

365 days, 

366 days, 

Also, 7 days make 1 
100 years 1 century. 



m. 



1 hour, 


h. 


1 day, 


d. 


1 conimo~ vear, 


c. y. 


1 leap yeai, 


i.y. 



week, 12 calendar months 1 year, and 



THE LONG TON FOR WEIGHING COAL. 

Formerly, 112 pounds, or 4 quarters of 2S pounds each, were 
reckoned a hundred-weight, and 2240 pounds a ton, now called 
the long ton. This is now seldom employed in this country, 
except at the mines for coal, or at the United States Custom- 
houses for goods imported from Great Britain, in which country 
such weight continues to be used. 



CALENDAR OF MONTHS AND DAYS IN A YEAR. 



January, 


1st mo. 


31 days. 


July, 


7th mo. 


31 days 


February, 


2d " 


28 or 29. 


August, 


8th " 


31 " 


March, 


3d « 


31 days. 


September, 


9th " 


30 " 


April, 


4th " 


30 " 


October, 


10th " 


31 " 


May, 


5 th " 


31 " 


November, 


11th « 


30 " 


June, 


6th " 


30 " 


December, 


12th " 


31 " 



MISCELLANEOUS MEASURES. 
Counting. Papek. 



12 units make 1 dozen. 

12 dozen, 1 gross. 

20 units, 1 score. 

5 scores, 1 hundred. 



24 sheets make 1 quire. 
20 quires, 1 ream. 

2 reams, 1 bundle. 

5 bundles, 1 bale. 



J 6 Maud JJook of Calculations. 



USEFUL NUMBERS FOR ENGINEERS. 

12 inches make one lineal foot. 144 square inches make one 
square foot. 1728 cubic inches make one cubic foot. 
6 feet in length=l fathom. 

CUBIC INCHES IN BUSHELS AND GALLONS. 

The standard bushel of the United Slates contains 2150.42 
uubic inches ; and the Imperial bushel of Great Britain, 
.2218.192 cubic inches. 

The standard liquid gallon of the United States contains 231 
cubic inches, and the Imperial gallon of Great Britain, 
277.274 cubic inches. The latter is used below. 

WEIGHTS OF WATER. 

One gallon of fresh water weighs 10 lbs. 

One gallon of sea water weighs 10£ lbs. 

One gallon=To 6 oth of a cubic foot. 

One cubic foot=6^ gallons. 

One cubic foot of fresh water weighs 62^ lbs. =1000 ozs. 

One cubic foot of sea water weighs 64 lbs. 

A CORD OF WOOD. 

A pile of wood 8 feet long, 4 feet wide, and 4 feet high, is ar 
cord. A cord foot (c. f.) is 1 foot in length of this pile, or 16 
cubic feet. 

MEASUREMENTS OF GOVERNMENT LAND. 

640 acres or 1 square mile make one section of land; 320 
acres=i section; 160 acres=i section. 

WEIGHTS OF METALS. 

Wrought Iron, 3dhr cubic inches=l lb., or leu. in.=. 2778 of alb. 

st Iron, 3 T fo " " =1 " " =.257 " 

el, soft, 3 T !o " " =1 " « =.2814 iS 



U 9 


t( 


3xto 


a 


3tBo 


a 



iss, 3roo " " =1 " " =.3 



Hand Book of Calculations. 37 

UNITED STATES MONEY. 

Table. 

10 mills are 1 cent,, ^ ct. 

10 cents are 1 dime, d. 

3 dimes or 100 cents are 1 dollar, dol. or $. 

10 dollars are 1 eagle, E. 

The dollar is the unit; hence dollars are written with the sign 
$ prefixed to them and the decimal point placed after them. 

Cents occupy hundredths place on the right of the decimal 
point and occupy two places, hence if the number to be expressed 
is less than 10 a cipher must be prefixed to the figure denoting 
them; one dollar and nine cents is written $1.09. 

Mills occupy the place of thousandths. In business calcula- 
tions, if the mills in the result are 5 or more, they are consid- 
ered a cent; if less than 5 they are omitted. 

STEELING OR ENGLISH MONEY. 
Table. 

4 farthings (qr. or far.) make 1 penny d. 

12 pence 1 shilling, s. 

20 shillings, 1 pound or sovereign £ 

10 florins (fl.) 1 pound, £ 

FRENCH MONEY. 

Table. 

10 centimes = 1 decime. 
10 decimes = 1 franc. 

The unit of French money is the franc, the value of which in 
TJ. S. money is 19.3 cents, or about i of a dollar. 

The Money Unit of the German Empire is the mark, which is 
divided into 100 pennies. The value of a mark is $0,238, or 
nearly Si- 

Canada money is expressed in dollars, cents and mills, which 
have the nominal value of the corresponding denominations of 
U. S. money. 



JS 



Hand Book of Calcnlatio7is. 



ROMAN TABLE. 



I. denotes 


One. 


XVII. denotes 


Seventeen. 


II. 


Two. 


XVIII. 


Eighteen. 


III. 


Three. 


XIX. 


Nineteen. 


IV. 


Four. 


XX. 


Twenty. 


V. 


Five. 


XXX. 


Thirty. 


VI. 


Six. 


XL. 


Fortv. 


VII. 


Seven. 


L. 


Fifty. 


VIII. 


Eight. 


LX. 


Sixty. 


IX. 


Nine. 


LXX. 


Seventy. 


X. 


Ten. 


LXXX. 


Eighty. 


XI. 


Eleven. 


XO. 


Ninety. 


XII. 


Twelve. 


0. 


One hundred. 


XIII. 


Thirteen. 


D. 


Five hundred, 


XIV. 


Fourteen. 


M. 


One thousand 


XV. 


Fifteen. 


X. 


Ten thousand 


XVI. 


Sixteen. 


M. 


One million. 



TABLE OF ALIQUOT PARTS. 



Of a 


$. 


Of a Ton. 


Of a cwt. 


Of an Acre. 


Of a Month. 


cts. 


% 


cwt. ton. 


lb. cwt. 


id. A. 


d. m. 


50 == 


i 

"2" 


10 = i 


50 = -J 


80 = i 


18 = i 


33* = 


* 


5 = i 


25 = i 


40 = i 


10 — i 


25 = 


i 


4 — . i 


20 == | 


32 = i 


li — 'i 


1» = 


i 


H= i 


m= i 


20 = i 


fi = i 


18*- 


i 


' 2 — T V 


™ = T v 


16 = T V 


5 = i 


10 = 


l 
TIT 


1 = 


5 = 1 


8 = A 


3 -A 



><"OTE. 



An Aliquot part of a number is an exact divisor of it; thus, 
2, 4 and 8 are exact divisors of 16. 



Hand Book of Calculations. jp 

MISCELLANEOUS MEASURES. 

3 inches = 1 palm. 3.28 Feet = 1 meter. 

4 " = 1 hand. 6 " — 1 fathom. 
6 " = 1 span. 830 Fathoms = 1 mile. 

18 " =1 cubit. 3 Knots = 1 marine league. 

21.8 " =1 Bible cubit. 60 Knots \ 

2^ Feet = 1 military pace. 69£ Statute miles > =1 degree. 

3 " = 1 common pace. 991.12 Miles ) 

¥ V of an inch=a hair's breadth. 

TABLE. 

Showing relative value of French and English measures of 
length. 

French. English. 

Milimeter, ... = .. 0.03037 inches. 

Contimetre, . . . = . . . 0.39371 " 

Decimetre, ... = .. 3.93710 " 

Metre, •... = .. .39.37100 <• 

In the French system of weights and measures, which has 
been legalized by special act of the U. S. Congress, the metre, 
litre, gramme, etc., are increased or decreased by the following 
words prefixed to them: 

Milli expresses the 1,000th part. 

Centi " " 100th " 

Deci " " 10th " 

Deca " 10 times the value. 

Hecato " 100 " " 

Chilio " 1,000 " " " 

Myrio " 10,000 " " << 

The following approximate measures, though not strictly ac- 
curate, are often useful in practical life. 
45 drops of water, or a common teaspoonful=l fluid drachm. 

A common tablespoonful=^ fluid ounce. 

A small teacupful, or 1 gill =4 fluid ounces. 

A pint of pure wate = l pound. 

4 tablespoon fuls, or a wine glass — -J gill. 

A common-3ized tumbler =-£ pint. 

4 teaspoonfuls = l tablespoonful. 



4Q Hand Book of Calculations. 



REDUCTION. 

Reduction is changing compound numbers from one denom- 
ination to another without altering their values. It is of two 
kinds, Descending and Ascending. 

Reduction Descending is changing higher denominations to 
lower, as tons to pounds. Reduction Ascending is changing 
lower to higher denominations as cents to dollars. 

To reduce higher denominations to lower. 

RUL3. 

Multiply the number of the highest denomination given, by 
the number required of the next lower denomination to make 
one of that higher, and to the product add the number, if any, 
of the lower denomination. 

Proceed in like manner till the whole is reduced to the re- 
quired denomination. 

Example ik Troy Weights. 
Reduce 63 lb. oz. 10 pwt., to pennyweights. 

OPERATION. 

63 lb. oz. 10 pwt. 

12 . ■■ " 



12ft 
63 



Since in 1 pound there are 12 ounces, 
in 63 pounds there are 63 times 12 
ounces, cr 7.6 ounces. 
75 g 0z Since in 1 ounce there are 20 penny- 

20 weights, in 756 ounces there are 756 

times 20 penny-weights: and 10 penny- 

15120 weights added, make 15130 penny- 

10 weights. 



15130 pwt., Ans. 



Hand Book, of Calculations. dr 

Example lx Avoirdupois Weight. 
Reduce six tons, eight hundred weight, three quarters to lbs* 



6 T. 8 ewt. 3 
20 


qrs. 


120 

8 add above. 




128 cwt. 
4 




512 
3 




515 qrs. 
,. 25 




2575 
1030 





12875 lbs. Answer. 

Examples for Practice. 

1. Reduce 116 tons 68 lbs. to ounces. 

2. Reduce 208 tons 42 lbs. to pounds. 

3. Reduce 180 degrees of the circle to seconds. 

4. Reduce 365 d. 5 h. 48 mi. 50. sec. to seconds. 

5. Reduce 75 b. 3 pk. 5 qt; to quarts. 

■ >'. 

Note. 

Expertness in this rule of arithmetic is of considerable im- 
portance, as it enters into a vast number of practical questions 
in every department of manufacturing as well as engineering. 



42 Hand Book of Calculations. 



To reduce lower denominations to higher. 

Rule. 

Divide the given number by the number of its denomination 
required to make one of the next higher, and reserve the re- 
mainder, if any. 

Proceed in like manner with the quotient, and so continue 
uutil the whole is reduced to the required denomination. 

The number of the required denomination, with the several 
remainders, if any, will be the answer. 

Examples. 

1. Bring 98,704,623 lbs. to tons and lbs. 

2000)98704623 



49352 Tons, 62a lbs. 
Bring 9876 lbs. coal to the long ton, cwt., qrs. and lbs. 
2240)987fi(4 tons. 
8960 

112)916(8 cwt. 
896 



28)20(0 qrs. 
Ans. 4 tons 8 cwts. qrs. 20 lbs. 

Pkoof. 

Reduction Ascending and Descending prove each other ; for 
one is the reverse of the other. 

Notes. 

A simple number is one which expresses one or more units 
of the same denomination. 

A compound number expresses units of two or more de- 
nominations of the same kind, as 5 yards, 1 foot, 4 inches — or 
example, page 41, 6 T., 8 cwt, 3 qrs.,— these are compound 
numbers ; but ten oxen, or five dollars, are simple numbers. 



Hand Boj/c of Calculations. 4.3 



Example. 



76,245 gills to gallons, etc. 
4)76245 



2)190'il— 1 gill. 



4)9530—1 pint. 

2382—2 quarts. 
Ans. 2382 gallons, 2 quarts, 1 pint and 1 gill. 

Examples for Exercise. 

1. In 76,298 ounces how many tons, etc. 

2. In 648,000 seconds how many degrees? 

3. In 15,130 pennyweights how many pounds, etc.? 

4. In 3,760,128 cubic inches how many cords? 

5. In 785 pints how many gallons ? 

EXAMPLES IN THE TABLE OF WAGES. 

1. What is the amount of 7 weeks, 4-J days work at 7 dollars 
per week. 

7 weeks. 
7 dollars. 

49=7 weeks pay. 
4.6 ; f=4 days per table. 
58J=5 hours or i day per table. 



54.25 Ans. Fifty-four dollars and 25 cents. 
2. What is the amount of one week and $ day extra time at 
$18.00 per week ? 

1 week = 18.00 
5 hours= 1.50 



19.50 Ans. 
3. What do a boy's wages come to, for 5-J days, at #5.00 per 
week ? 

5 days = 4. 16§ per table. 
5 hours = 41 J per table. 



4.684- Ans. $4.68. 



44 



Hand Book of Calculations. 



4. What do 46 days, 6 hours and a quarter, amount to at 
$17.00 per week. 

1 day per table = 2. 83 J 
multiply by ( X ) 46 days. 











1 5^ amount of fraction. 
1698 


1 hom 


r28j * 
byi 
7A, 


» 




1132 




divide 


13033^=46 days. 

170 =6 hours per table. 
7iV=i hour. 




13210 T 5 2 Ans. $132.10. 








Examples for Practice. 




5. 


Howm 


uch 4 


days 


\ 2i hrs. at $ 8. 00 per week. 


Ans. 15.67 


6. 




306 


a 


" " 9.00 " " 


" 459.00 


7. 




184 


a 


5 " " 11.00 " « 


" 33825 


8. 




11 


a 


" " 4.00 " " 


ff 


9. 




39 


a 


6 " li 15.00 " " 


ff 


10. 




1 


i( 


2 « " 12.00 " " 


ft 


11. 








Hi" " 14.00 " " 


a 


12. 








5 " " 7.00 " " 


tt 



Example. 
In doing the sum for example 9, do it like this: 
1 day at $15.00 per week is $2.50 
Multiply by 40 days, 



Deduct 4 hours at 25c., 



40 

100.00 
1.00 



Answer, $99.00 

There are various " short cuts " in figuring wages, like the 
last example, which it is well to become familiar with, so that 
in this important part of mathematics, both quickness and 
accuracy may be attained. 

Note. 

When the fraction is less than i cent it is the gain of the 
employer by the amount of the fraction — but, if the fraction is 
more (like f ) it is called a full cent and goes as a full cent to 
the employee. 



Hand Book of Calculations. 4.5 



NATURAL OR MECHANICAL 
PHILOSOPHY. 



Natural philosophy is the science which treats of the laws of 
the material world; and it is this science, with which the engi- 
neer has to cooperate, in obtaining the best results from his 
professional skill. All the calculations relating to steam-engi- 
neering, are closely connected with the operations set forth in 
that department of knowledge which is thus termed. 

u I have learned more about my business/' said a trusted and 
competent engineer, to the author " from an old work on nat- 
ural philosophy, which I own, than from all the other books I 
ever read." Hence it is worth the while, to consider a little, the 
foundation of this important part of an engineer's education. 

Natural or Mechanical philosophy is divided into Mechanics, 
Hydrostics, Pneumatics and Electricity; the engineer in his 
daily practice is liable to be called upon to deal with one or all 
of them, for he has to do with machinery, treated under the 
head first named; with water, treated under the division, 
Irostatics; air (Pneumatics) and with Electricity; upon 
analysis it will appear that all the computations in this volume 
are practically used, in connection with one or more of these 
divisions. 



jf6 Hand Book of Calculations. 

Science shows that there are but few fundamental laws be- 
neath all creation, and all observation proves that these basis 
principles are preserved through countless varying forms, 
therefore, 

Let it be particularly noted that there are but 68 elemen- 
tary substances, known at the present day, to exist; these are 
platinum, gold, silver, copper, iron, lead, tin, sulphur, nickel, 
mercury, carbon, hydrogen, nitrogen, antimony, arsenic, bis- 
muth, etc., etc. A substance which cannot be resolved into 
two or more different substances is called an elementary or sim- 
ple body; as for example, neither water, coal, nor brass are 
elementary substances as each can be resolved into other forms, 
cf matter. 

Matter is any collection of substance existing by itself in a 
separate form. Matter appears to us in various shapes, which 
however can all be reduced to two classes, namely solids or 
fluids. 

A Solid offers resistance both to change of shape, and to 
change of bulk. 

A Fluid is a body which offers no resistance to change of 
shape. 

Fluids again, can be divided into liquids and vapors or gases. 
Water is the most familiar example of a liquid. A liquid can 
be poured in drops while a gas or vapor cannot. It is import- 
ant to note that experiment proves that every vapor becomes a 
gas at a sufficiently high temperature and low pressure, and, on 
the other hand, every gas becomes a vapor, at sufficiently low 
and high pressure. 

Atoms. An atom is the smallest particle of matter known 
to exist, they are sometimes called molecules, and are so small 
that they cannot be divided. 

Chemistry treats of all which relates to these particles of 
matter, and to the changes of constitution produced by their 
action on each other. The combustion of coal is strictly a 
chemical process, as the mass of fuel is reduced to particles of 



Hand Book of Calculations. 47 

gas and vapor by combination with oxygen, resulting in heat,, 
which in turn expands water into steam, in the boiler. 

Now we are brought, with our 68 original elementary sub- 
stances, to those forces which act upon them, live in number ; 
these will be explained in the next section, under the title 
of primary powers. 



PRIMARY POWERS. 



The following is a list of all the primary powers which, as 
yet, have been used by man in accomplishing his purpose in the 
wide domain of practical life. These are 

1. Water power. 
2. Wind power. 
3. Tide power. 



lide power. 
4. The power of combustion. 
5. The power of vital action. 



To this list may hereafter be added the power of the volcano 
and the internal heat of the earth; and besides these, science at 
the present time gives no evidence of any other 

Gravitation, electricity, galvanism, magnetism and chemical 
affinity can never be employed as original sources of power. 
There is no more prevalent and mischievous error than to sup- 
pose that work can be had from these latter, and no engineer of 
intelligence will waste his life energy in trying to get "some- 
thing from nothing " as he will be doing should he attempt the 
problem. 

Even in the modern application of electricity it is apparent 
that it is but the resovoir (a storage battery) or the means of 
transfer by wires, of the power of combustion, or water, to the 
work. 

The same must be said of the elastic force of steam, of air 



^S Hand Book of Calculations. 

and of springs; and also of machinery; they are all bnt the act- 
ive agents employed between the primary p6 to er and the work. 

In ail computations of power and the action of machines 
these first principles should always be borne in mind; it is not 
the engine which is the source of motion to the machinery, nor 
yet the steam, but the repulsive energy imparted to the expand- 
ing water from the burning fuel. 



THE MECHANICAL POWERS. 



We now proceed to consider the effect produced, when these 
forces are made to act by the intervention of other bodies. 
These intermediate bodies are called machines and by the means 
of them the effect of a given force may be increased or dimin- 
ished as desired. 

Machines ?re divided into simple and compound. The simple 
machines or what are commonly called Mechanical Powees, 
are six in number; viz. i 

1. The lever. 
2. The wheel and axle. 
3. The pulley. 

4. The inclined plane. 
5. The screw. 
6. The wedge. 

These can in turn be reduced to three classes: 

I. A solid body turning on an axis. 
II. A flexible cord. . 
III. A hard and smooth inclined surface. 

For the mechanism of the wheel and axle and of the pulley, 
merely combines the principle of the lever with the tension of 
the cords; the properties of the screw depend entirely on those 
of the lever and the inclined plane; and the case of the wedge 
is analogous to that of a body sustained between two inclined 
planes. 



Hand Book of Calculations. 



49 



MACHINERY. 



Compound machines are formed from two or more simple 
machines. Tools are the simplest implements of art; these 
when they become complicated in their structure become ma- 
chines, and machines when they act with great power, take the 
name, generally speaking, of engines. 

The advantage that man has gained by pressing into his ser- 
vice the great forces of nature, instead of depending on his own 
feeble arm. is evinced by the fact that aided by the steam 
engine one man can now accomplish as much labor as 27,000 
Egyptians, working at the rate at which they built the pyra- 
mids (Dapin). 

The mechanical powers will now be separately considered, it 
being remembered that none of them create force, but that 
they only modify and direct it, acting by certain great laws, 
established by the supreme Creator and generous Giver of the 
original sources, of both the Primary and Mechanical causes. 
He will labor most effectively and happily who studies these 
laws and acts in accordance with their principles, which are 
those laid down and explained in detail in books relating to 
N'atural Philosophy. 

THE LEVER. 




Lever first kind. 



jo Hand Book of Calculations. 

THE LEVEK. 

The lever is an inflexible bar or rod, some point of which 
being supported, the rod itself is movable freely about that 
point as a center of motion. 

This center of motion is called the Fulcrum or Prop. 

In the lever three points are to be considered, viz. : the ful- 
crum or point about which the bar turns, the point where the 
force is applied, and the point where the weight is applied. 

There are three varieties of the lever, according as the ful- 
cram, the weight or the power is placed between the other two, 
but the action in every case is reducible to the same principle 
and the same general rule applies to them all. 

Note. 

When two forces act on each other by means of any machine,, 
that which gives it motion is called the power, that which 
receives it th b weight, hence, 

In the diagrams the letter P is used to denote the point of 
application of the forces ; the letter F denotes the fulcrum, 
or prop, and W the weight. 

1st. When the fulcrum (F) is between the force (P) and 
the weight (W). Fig. 1. 




Fig. 2. Lever 2d kind. 

2d. When the weight (W) is between the fulcrum (F) and 
the force (P). Fig 2. 



Hand Book of Calculations. 



5* 



THE LEVER. 



C 



w 



* 



j^ 
s 




\ 



Fig. 3. Lever 3rd kind. 



3rd. When the force (P) is between the fulcrum (F) and 
the weight (W). Fig. 3. 

General Eule. 

The force (P) multiplied by its distance from the fulcrum 
(F) is equal to the weight (W) multiplied by its distance from 
the fulcrum. 

In the following examples the distances are figured in inches 
and the weight in pounds, the unit of distance in mechanics 
being one inch, and the unit of loeight being one pound. 

Note. 

The following calculations are made on the supposition that 
the action of the mechanical powers is not impeded by their 
own weight, or by friction and resistance. Thus, in each cal- 
culation, in figuring the problems relating to the safety-valve, 
the weight of the valve, spindle and lever have to be taken into 
the estimate. A special rule (with illustrations) will be. given 
in its proper place to show how these are to be provided for. 

Example. 



What force applied at three feet from the fulcrum will bal- 
ance a weight of 112 lbs. applied at 6 inches from the fulcrum 
(observe diagram of 1st form of lever). Here the leverages are 
36 and 6 inches. 



§2 Hand Book of Calculations. 

THE LEVER. 
This is found by dividing 672 by 36. 



112 lbs. 
6 inches. 

36)672(18f 
36 


Pkoof. 
112 lbs. X 6 inches = 672. 
18f " X 36 " = 672. 


312 

288 




24 

9. 




36 





That is, 18f lbs. applied at the end of a 3 J foot bar with a 
fulcrum 6 inches from the point, will lift a box weighing 112 
lbs. 

Example. 

If 80 lbs. be applied at the extreme end of a 5 foot lever (with 
prop 1 foot from the point), what force is needed to balance the 
80 lbs. The two leverages being 48 inches and 12 inches. 

Now, multiply the force (P) 80 lbs., by the distance from 
the fulcrum (F) 48 inches and divide by 12 inches. 

48 inches. Proof. 
80 lbs. 48x80 lbs. =3840 
12X320 "=3840 



12 in.)3840 



320 lbs. 
This is an example worked from the lever of the second kind. 



Hand Book of Calculations, 



53 



THE LEVER. 



Under the general rule given, it will be seen that under all 
circumstances the force multiplied by its distance from the ful- 
crum, is equal to — or balanced by, the weight multiplied by its 
distance from the fulcrum; 4 sub-rules are added which will 
cover all problems where only three of the numbers are known. 




Fig. 4. Lever 1st kind. 

To find the power (P) on any lever, ivhen the weight (W) and 
two distances from the fulcrum (b)are given. 

Sub-Rule 1. 

Multiply the weight (W) by its distance from the fulcrum 
(b) and divide by the distance from P, to b. 

The quotient is the power. 



Example. 

How much to balance 200 lbs., 18 inches from the fulcrum 
(b) to the end of the lever at (P). The whole length of the 
lever being 36 inches. 
18 in. 
200 lbs. 



36 in.)3600(100 lbs. Answer. 

The exMKiple given to illustrate the general rule is similar to 
this. 



54 



Hand Book of Calculations. 



THE LEVER. 




Fig. 5. Lever of the 2d kind. 

To find the weight (W) when the power (P) and the two dis- 
tances from the fulcrum (b) are given. 

Sub-Rule 2. 

Multiply the power (P) by its distance from the fulcrum (b) 
and divide by the distance of the weight (W) from the fulcrum. 
The quotient is the weight. 

Example. 

If 480 lbs. be applied at the end of a lever, 135 inches from 
the fulcrum, what weight will it lift 45 inches distance from the 
fulcrum. 



480 


Prooe. 


135 


±440 x 45 = 64,800 





480x135 = 64,800. 


2400 




1440 




480 




)64800(14 


40 lbs. Ans. 


45 




198 




180 




180 


fp>_ 


180 





Hand Book of Calculations. 55 



THE LEVER. 




Z 3 

Fig. 6. Lever of the 3rd kind. 

To find the distance of the poiver (P) from the fulcrum (b) 
the weight and its distance and the power being given. 

Sub-Rule 3. 

Multiply the weight (W) by its distance from the fulcrum 
tad divide by the power. 

Example. 

If a weight 900 lbs. be 12 inches from the fulcrum, at what 
^stance must 80 lbs. be placed to balance it ? 

12 Proof. 

900 900 12 10,800. 

135 80 10,800. 

80)10800(135 inches Ans. 
80 

280 
240 

400 

To find the distance of the weight from the fulcrum. The 
power and its distance from the fulcrum and toeight being 
known. 

Sub-Rule 4. 

Multiply the power by its distance from the fulcrum and 
divide by the weight. 



$6 Hand Book of Calculations. 



THE LEVER. 

Example. 

(Lever 3d kind.) If the power be 1,000 lbs., 3 inches from 
the fulcrum, at what distance must the weight (W) 120 lbs. be 
placed to balance it. 

1,000 lbs. power. Proof. 

3 in. distance. 1,000 3 = 3,000. 

120 25 = 3,000. 



120)3000 



25 inches. 

Ans. 25 inches from the fulcrum. 

The Leverage of the Power. 

The ratio of the power end of the lever, to the length of the 
weight end, is called the leverage of the poioer. 

The three varieties of the lever are shown in Fig. 4, 5 and 6, 
and in each case the lever is supposed to be seven feet long, 
and divided into feet. 

The respective lengths (fig. 4) being 6 feet and 1 foot, the 
leverage is 6 to 1, or 6. In the second (fig. 5) it is 7 to 1, 
or 7 ; in the third one-seventh to 1, or 1-7, showing that in 
the first case the power balances 6 times its own amount ; 
in the second case 7 times its amount ; in the third case only 
one-seventh of itself, because it is nearer the fulcrum than the 
weight. 



Hand Book of Calculations. 



57 



THE WHEEL AND AXLE, or PERPETUAL LEVER. 




Fig. 7. 

When a lever is applied to raise a weight, or to overcome a 
resistance, the space through which it acts at one time is small 
and the work mast be accomplished by a succession of short and 
intermitting efforts. The common lever is, therefore, used 
only in cases where weights are required to be raised through 
short spaces. When a continuous motion is required, as in rais- 
ing ore from the mine, or in weighing the anchor of a vessel, 
some contrivance must be adopted to remove the intermitting 
action of the lever and render- it continuous. The wheel and 
axle, in its various forms, fully answers this purpose. It may 
be considered a revolving lever. 

The wheel and axle may be likened, also, to a couple of 
pullies of different diameters united together on one axis, of 
which the larger is the wheel and the smaller the axle, with, a 
common fulcrum. 

The power of the wheel and axle is expressed by the number 
of times the diameter of the axle is contained in that of the 
wheel, as per the following 



Rule. 



Multiply the power at the edge of the wheel by its radius 
(half its diameter) and divide the product by the radius of the 
axle. The quotient is the weight that the power will raise. 



jS Hand Book of Calculations. 

THE WHEEL AND AXLE. 

Example. 

Required the weight that can be raised by a power of 50 
lbs. applied at the circumference of a wheel of 5 feet diameter 
(2^ ft. radius) the weight to be attached to the end of a rope, 
which is to be wound around a barrel or axle 12 inches in diam- 
eter. Now then 

%i feet = 30 inches. 

50 lbs power. 



Radius of axle 6)1500 

250 lbs. answer. 

Note. 

There are obviously two ways by which the power of the 
wheel and axle may be increased; either by increasing the di- 
ameter of the wheel or diminishing that of the axle. 

The weight to le raised, the diameter of the axle and dameter 
of the wheel being given, to find the amount of poiver required 
to raise the weight. 

Rule. 

Multiply the weight to he raised by the radius of the axle, 
and divide the product by the radius of the wheel. 

Example. 

Required the power necessary to raise a weight of 400 lbs. by 
an axle of 10 inches, and wheel of 50 inches in diameter. 
.Now, then: 

Weight — 400 
■J diam. of axle 5 



i diam. of wheel 25)2000(80 lbs. Ans. 
2000 



Hand Book of Calculations. 59 



THE WHEEL AND AXLE. 
A ship's capstan is another form of the wheel and axle. 

Example for Practice. 

In weighing anchor 6 capstan ^...... „_X^ ^ t~Ls 

bars are used ; from center of ^^sg^n^^a^ 

capstan to point of pressure is L [ 

6 feet; diameter of axle of cap- W^^S±==^ 

stan = 24 inches. Now then, /// 1 (^ Oi\\\ 

if each man exerts 80 lbs. with / 1 1 / 7 ^ ^ a\\\ 

his bar. //////! II! \W 



The leverage for force (radius of 12 ft. diam.}=6. 

Number of men 6 



36 
Lbs. for each man 80 



Divide by radius of axle 1)2880 



2880 lbs. Ans. 

If an allowance of ten per cent, is made for friction and the 
rigidity of the cord, the answer will be 2592 lbs. Ans. 

Example for Practice. 

The diameter of a steering wheel on a ship is 5 feet and the 
barrel is 15 inches in diameter. If a man appjies a force equal 
to 200 lbs. what resistance would he overcome? Ans. 800 lbs. 

THE CHINESE WHEEL AND AXLE. 

To combine the requisite strength with moderate dimensions 
and great mechanical powtr has been accomplished by giving 
different thicknesses to different parts of the axle and carrying 



6o 



Hand Book of Calculations. 



THE WHEEL AND AXLE. 




Fig. 9. 

a rope which is coiled on the inner part through a pulley at- 
tached to the weight and coiling it in the opposite direction on 
the thicker part as in fig 9. 

We see h:re exemplified the principle, that the weight sus- 
tained by a given power, may be increased as its velocity is di- 
minished. By inspecting fig. 9 it will be seen that the rope 
connected with the thinner part of the axle unwinds, while that 
connected with the thicker part winds up, by which means the 
ascent of the weight may be rendered slow in any degree, and a> 
proportionally greater quantity of matter may be added. 

To find power in this arrangement follow the 

Rule. 

The power multiplied by the radius of the wheel, in feet, is- 
equal to half the weight multiplied by the difference in the= 
half diameters (radii) of the thicker and thinner parts of the 
axle. This will be made clear by the following 

Example. 



The diameters in fig. 9 are 1 foot and f of a foot; the length 
of the handle 2 feet 3 inches ; if the exertion put forth is equal 
to 80 lbs. what weight will be lifted. 

. Now then to follow the rule. 



Hand Book of Calculations. 61 



THE WHEEL AND AXLE. 

The length of the handle 2,25 feet. 
The power exerted 80 lbs. 



i the difference in the radii .0625)18000(2880 lbs. Ans. 

1250 



5500 
5000 

5000 
5000 



In all these examples the diameter of the rope has been sup- 
posed to be so small in comparison to that of the drum or bar- 
rel that it has been neglected; if it is a thick rope, then the 
leverage must be measured from the center of the barrel to the 
center of the rope. 

Example. 

Wheel and axle, the barrel is 10" in diameter, the rope is 1-j- 
inches in diameter, the crank handle is 15" radius, and the 
weight to be lifted is 500. What force must be applied to the 
handle if 10 per cent, is to be added for friction. Now, then 

Leverage of weight 5"-f-f " = 575. Being radius of barrel 
and rope. 

500X5,75= 

500 lbs. 



15)287500(191| lbs. Ans. 
15 



137 
135 



25 
15 



10 » 

Add for friction 19.17 = 210 ,Vo Ans. 



62 



Hand Book of Calculations. 



THE WHEEL AND AXLE. 

These examples are worked in decimal fractions, the rules 
and examples of which will be given later. 

To find the difference in the half diameter of the axle, (Fig. 
9.) proceed thus : 1 foot — f = \ foot; \ this for the radius = 
one-eighth foot, and half this is one-sixteenth, or in deci- 
mals .0625. (See example.) 

THE PULLEY. 

The pulley is a wheel over 
which a cord, or chain or band 
is passed, in order to. transmit 
the force applied to the cord 
in another direction. The 
practical effect of the machine 
depends upon the rope, the 
wheel being introduced to 
diminish friction and the effect 
of imperfect flexibility, but the 
whole effect of imperfect flexi- 
bility and friction are not de- 
stroyed, although in calcula- 
tions, we proceed as though they 
were. 

There is no mechanical advantage gained by a single rope 
over one or more fixed p allies; but this combination is of the 
greatest use by enabling us to change the direction of the force. 

Pulleys are divided into fixed and movable. In the fixed 
pulley no mechanical advantage is gained, as already explained, 
but its use is of the greatest importance in accomplishing the 
work appropriate to the pulley, such as raising water from a 
well. Thus, it is far more convenient to raise a bucket from a 
well by drawing downward, as is the case where the rope passes 
over a fixed pulley above the head, than by drawing upward 
leaning over the curbing. 

From its portable form, its cheapness and the facility 
with which it can be applied, especially in changing or modify- 
ing the direction of motion, the pulley is one of the most con- 
venient and useful of the mechanical powers. 




Fig.. 10. 



Hand Book of CaI.cn/atio7is. 



6j 



THE PULLEY. 
It must be observed that in using any system of movable 
pullies, the whole weight of the pulleys themselves, together 
with the resistance occasioned by the friction and rigidity of the 
ropes all act against the power and so far lesseu the weight 
which it is capable of raising. 

The moveable pulley by distributing the weights into separate 
parts, is attended by mechanical advantages proportioned to the 
number of points of support. Movable pulleys may be arranged 
according to different system's which increase the efficacy of a 
given power in different degrees. 

By means of the pulley great 
facilities are afforded in raising 
heavy weights, as boxes of mer- 
chandise or heavy blocks of 
stone. Fig. 1 1 represents a con- 
venient method in building- 
brick chimneys for steam plants 
which has been observed by the 
author, as used by Glasgow, 
Scotland, masons and builders. 
The crane at B enables the 
workmen when the brick and 
mortar are raised, to swing it 
around to the point where it is to 
Fig. 11. be laid or to a platform near it. 

The lower cord of the rope C D is connected with a wheel and 
axle ; in the illustration, it may be seen, that instead of the 
wheel and axle we might fasten a horse to the rope, or attach a 
sweep to the top of the axis and join a team of horses to the 
end of it to expedite the work. 

The employment of this device, in sufficiently large chimneys, 
enables the builder to dispense with the use of scaffolding, the 
workmen building into the corners of the chimney, as the work 
progresses, a ladder of \ or \ inch round iron every fifteen inches, 
to enable them to go up and down in the interior of the flue. 
Tli us a large expense is saved in cost of scaffold, and the risk is 
less for the mason. 




6 4 



Hand Book of Calculations. 



THE PULLEY. 

Fast and loose pullies. These are shown 
in Fig. 12 where the movable block A car- 
ries the weight with a fixed counterpart 
B. Here the rope is attached by one end 
to the fixed block and is passed over the 
movable and fixed pullies from one to the 
other in succession, the power being ap- 
plied to the other end. This system is 
known as fast and loose pulley blocks. 

The fixed end of the rope is sometimes 
fastened to the movable block. 

To find the power necessary to balance 
the weight by the means of a system of 
fast and loose pulleys. 

EtJLE 1. 

Divide the weight by the number of 
ropes by which it is carried; that is by 
Fig. 1 2. the number of ropes which proceed from 

the movable block. The quotient is the power required to bal- 
ance the weight. 

Example. 
A cylinder cover weighing 1200 lbs. is lifted by a pair of 
blocks of two sheaves each, the rope is fastened to the upper 
block. 
Now, then, in two sheaves there are 4 ropes. 
4)1200 




300 lbs. will balance the weight of the cylinder 
head. 

Example. 
A boiler weighing 6 tons has to be lifted by a pair of treble 
blocks; how much power must be applied at the end of the 
rope to balance the weight. 

6 tons 
2000 



6)12000 lbs. 



2000 lbs. 



Hand Book of Calculations. 



65 




THE PULLEY 



Sometimes the upper block has 
4 sheaves and the lower 3, the 
rope being- r0 ve and fastened to 
the lower block, then when the 
stress comes, there will be 7 
singles of the rope holding the 
weight up. 

In this case the weight would 
be divided by 7. 

In all the above cases, single 
rope and a single movable block 
have been used, but we may have 
several movable blocks each with 
its own rope. 



Example. 

Let a pulley be fastened to a weight of 1200 lbs. and a rope 
fastened by one end to a beam, brought round the pulley, and 
the other end fastened to a second pulley; let a second rope be 
fastened to the beam, brought around this second pulley and 
fastened to a third pulley; let a third rope be fastened to the 
beam, brought round the third pulley and then up over a fixed 
pulley: what weight would put it in balance ? 

Axswer. 
In this case the 1200 lbs. is supported by two singles the first 
rope, hence each single rope bears a weight of 600 lbs. 

This 600 is the weight the second movable pulley sustains; 
bence each single of the second rope bears a strain of ™<>=r300 
lbs. 

This 300 lbs. is the weight the third movable pulley sustains; 
hence each single of the third rope bears '•',:" --150 lbs. Answer. 

In order to have the rules apply it is accessary to have the 
cords parallel with each other, as an} other than a "straight 
pull " altera the mechanical efficiency. 



66 



Hand Book of Calculations. 




THE PULLEY. 

The single fixed pulley as shown in 
Fig. 10 acts like a lever of the first hind, 
and simply changes the direction of the 
forces without modifying the intensity of 
the power. 

But the pulley may be employed as a 
lever of the second hind by suspending" 
the weight to the axis of the pulley, and 
fixing one end of the cord to a spot as a 
fulcrum point X as shown in Fig. 14. 
Thus the power acts through the diam- 
eter, A C B, in which B is the fulcrum. 



In acting as a lever of the third hind, 
the power is applied to the axis a in 
Fig. 15, one end of the cord being fixed 
at h and the weight attached to the 
other end, c. 

In the last example the gain is -|. 




Fig. 15. 




Hand Book of Calculations. 6j 



THE INCLINED PLANE. 

The inclined plane is a slope, or a flat surface inclined to the 
horizon, on which weights may be raised. By such substitu- 
tion of a sloping path for a direct upward line of ascent, a given 
weight can be raised by a power less than itself. 

The inclined plane becomes a mechanical power in conse- 
quence of its supporting part of the weight, and of course 
leaving only a part to be supported by the power. Thus the 
power has to encounter only a portion of the force of gravity at 
a time; a portion which is greater or less according as the plane 
is more or less elevated. 

The simplest example we have of the application of the 
inclined plane is that of a plank raised at the hinder end of a 
cart for the purpose of rolling in heavy articles, as barrels or 
hogsheads. Again, for another 

Example. 

When a horse is (hawing a heavy load on a perfectly hori- 
zontal plane, his force is spent chiefly in overcoming friction, 
and the resistance of the air, as the force of gravitation can 
afford no resistance, in the direction in which the load is 
moving. 

lint when the horse is drawing ;i load up a hill he has not 
only these impediments to overcome hut he lifts a part of the 
load. If the rise is 1 tool in 20, he lifts one twentieth of the 
load: if the ascent is one todi in four and the load is two tons, 
including his own weight, he Lifts 

4)4(10(1 



lone lbs. 



68 



Hand Book of Calciilations. 



THE INCLINED PLANE. 

Note. 

The general principle for all calculations relating to the 
inclined plane may be thus stated. As the length of the plane 
is to the height or angle of inclination, so is the weight to the 
power: this principle will be understood by reference to that 
part of this work relating to Ratio and Proportion. 





klfr*-* 










A 


r^^ 










r-n 




^-2^. 


"""*->. 






h 








*'*'••-. 




4- 










■■■■■■> 



Fig. 16. 

There are three elements of calculation in the inclined plane, 
the plane itself, the base or horizontal length and the height 
or vertical rise, together forming a right angled triangle. Fig. 
16 exhibits an inclined plane. 

To find the power necessary to raise a given weight, the 
length and heighth of the inclined plane being known. 

Rule. 

Multiply the weight by the height and divide by the length 
of the plane. 

Example. 

Required the power necessary to raise 1280 lbs. up an 
inclined plane 8 feet long and 5 feet high. Now then : 

1280 lbs. 
5 feet. 



8)6400 



800 lbs. Answer. 



Hand Book of Calculations. 



6 9 



THE INCLINED PLANE. 

The length and height of an inclined plane being known* to 
find the weight that a given power will support upon the plane, 

EULE. 

Multiply the power by the length of the plane and divide 
the product by the height. The quotient is the weight that 
the power will supp'vfc. 

Example. 

The length of an 'nclined plane is 15 feet; the perpendicular 
height 6 feet: what force will be required to sustain a weignt 
of 150 lbs.? F f g 16. 

150 lbs. 
6 feet. 

15)900(60 lbs. Answer. 
90 



00 




Fig. 17. 



The principle of the lever as applied to the inclined plane 
may be seen illustrated in Fig. 17, where the power is applied 
at the end of a cord passed round and over the weight (W ). 

In this case there is the action of a movable pulley, com- 
bined with an inclined plane, the rolling weight moved by a 
cord, B P, lapped round it, representing a movable pulley with 
the weight attached to the axle. Thus the leverage of the 
power on the inclined plane can be doubled. 



7° 



Hand Book of Calculations. 



THE SCREW. 




Fig. 18. 



The screw is an inclined plane wrapped around a cylinder. 

Take for example an inclined plane A, B, 0, Fig. 18, and 
bend it into a circular form resting on its base, so that 
the ends meet. The incline may be continued winding up- 
wards rcund the same axis and thus winding inclined planes of 
any length or height may be constructed. 

The distance apart of two consecutive coils, measured from 
centre to centre, or from upper side to upper side, (literally the 
height of the inclined plane), for one revolution, is "the pitch" 
of the screw. 

The screw is generally employed when severe pressure is to be 
exerted through small spaces; being subject to great loss from 
friction it usually exerts but a small power of itself, but derives 
its principal efficacy from the lever or wheel work with which 
it i-s very easily combined. 

A screw in one revolution will descend a distance equalto its 
pitch, or the distance between two threads and the force ap- 
plied to the screw will move through, in the same time the 
circumference of a circle whose diameter is twice the length of 
the lever. Hence the Rule. 



Hand Book of Calculations. 



7' 




Fig. 19. 



Kile. 



The power multiplied by the circumference is equal to the 
weight multiplied by the pitch. 

Example. 

If the distance between the threads be \ inch and the force 
of 100 lbs. be applied at the end of a lever 3 feet in length; 
what weight will be moved by the screw ? See the diagram 
Pig. 19. 

Twice the length of the lever =6 feet = 72 inches diam. 

100 power. 



7200 
3.14 to get circum. 



28800 
7200 
21000 



divide by pitch 5)22608.00 

.25 

5)452160 



00432 AriR. in lbs. 



J2 Hand Book of Calculations. 



THE SCREW. 

If the pitch of screw and length of lever be given, tvhat poiver 
will be required to move a given weight. 

Eule. 

The power multiplied by the circumference is equal to the 
weight multiplied by the pitch of screw. 

Example. 

If the pitch be f of an inch and the lever 2 feet, how much 
power must be applied at the end of the lever to raise a weight 
of 6 tons. 

6 tons= 12000 lbs. Xf 
3 



4)36000 
2 x 2=diam. X 12=48 



inchesX314=15072)90o00(59f lbs. 
75360 



146400 
135648 

10752 



Note. 

Out or Fig. 20 gives a view of the winding path of the 
endless screw. 




Fig. 20. 



Hand Book of Calculations. yj 



THE WEDGE. 



The wedge is a pair of inclined planes united by their basest 
or back to back. 



The wedge has a great advantage oyer all other mechanical 
powers in consequence of the way in which the power is applied 
to it, namely, by percussion, or a stroke, so that by the blow of 
a hammer or sledge almost any constant pressure is overcome. 

If instead of moving a load on an inclined plane, the plane 
itself is moved beneath the load, it then becomes a wedge. 
All cutting and piercing instruments, such as knives, razors,, 
scissors, chisels, nails, pins, needles, are wedges. 

The use of the wedge is to separate two bodies by force or to 
divide into two a single body. In some cases the wedge is 
moved by blows; in others it is moved by pressure. The action 
by simple pressure is to be considered. 

If the weight rests on a horizontal plane and a wedge be 
forced under it, when the wedge has penetrated its length, the 
weight will be lifted a height equal to the thickness of the butt 
end of the wedge, hence the Itule. 



yd Hand Book of Calculations. 

THE WEDGE. 

KlJLE. 

The power is equal to the weight, multiplied by the thickness 
of the wedge, divided by the length of the wedge. 

Example. 

A wedge 18 inches in length and 3 inches thick, is employed 
to lift a weight of 100 lbs. ;. what-pressure must be used ? 



< -- - T^ • - > • 

Fig. 21. 

Now then—The weight = 100 lbs. 

thickness = 3 inches. 

divided by 18)300(16§- lbs. Ans. 
18 

120 
108 

12 

18 

If a wedge be 12 inches long and 3 inches thick, and the 
pressure employed be 100 lbs., what weight will be lifted. 
This is the method of figuring: 

The power = 100 
Thelength= 12 



thickness 3)1200 



400 lbs. Ans. 



Hand Book of Calculations. 



75 



THE WEDGE. 

The wedge is generally formed of either 
wood or metal introduced into a cleft 
already made to receive it, as shown in 
Fig. 2->. 

Wlien two bodies are forced from one 
another by means of a ivedge. 

Rule. 

Multiply the resisting power by half the 
thickness of the head or back of the wedge, 
and divide the product by the length of 
one of its inclined sides. 



Example. 




Fig. 22. 



The thickness of the back of a double wedge is 6 inches, and 
its length, through the middle is 10 inches: what is the power 
necessary to separate a substance having a resistance of 150 lbs. ? 
Now then: 

150 lbs. to be overcome. 
-J- thickness 3 

10)450(45 lbs. Ans. 

In many cases, the utility of the wedge depends upon that 
which is entirely omitted in the theory, viz. the friction which 
arises between its surface and the substance which it divides — 
as in the case of nails, etc. The power generally acts by suc- 
cessive blows, and is therefore'subject to constant intermission, 
and but for the friction, the wedge would recoil between the 
intervals of the blows, with as much force as it had been driven 
forward, and the object of the labor would be constantly frus- 
trated. 



The rules for calculation do not apply to instances like the 
last described. 



76 



Hand Book of Calculations. 



SIZES, STRENGTH, ETC., OE ROPE. 



By reference to page 48 it will be observed that one of the 
three classes to which the mechanical powers may be reduced 
is that of & flexible cord; another name for the cord, or rope, is 
the funicular machine. Hence, the rules and calculations 
relating to ropes when used for the purpose of producing power, 
belong with those relating to the inclined plane and wheel and 
axle. 

Tic size of a rope is designated by the circumference meas- 
ured with a thread; thus a three inch rope measures three 
inches round. 

Ropes are made of iron, steel, manila and hemp, all of 
which, even of the same size, vary greatly in strength, durabil- 
ity and safety. 

All the tables given for strength of rope must be more or less 
modified by the time of service, the quality of material and 
method of manufacture; the strength of pieces from the same 
coil may vary one-quarter, and a few months service weakens 
rope from 20 to 50 per cent. A difference in the quality of 
hemp may also produce a difference of J in the strength of 
rope of the same size . 

Table. 

Shoiving what weight a hemp rope will bear hi safety. 



Circumfer- 
ence. 


Pounds. 


Circumfer- 
ence. 


Pounds. 


Circumfer- 
ence. 


Pounds. 


1 in. 


200 


3| 


2450 


64 


6050 


H'" 


312 


3| 


2*12 


51 


6612 


H 


612 


•4 


3200 


6 


7200 


2 


800 


4i 


4512 


6i 


7812 


n , 


10L2 


U 


4050 


6* 


8450 


n 


1250 


4f 


4512 


6f 


9112 


3 


1800 


5 


5000 


7 


9800 


3i 


2111 ! 


5i 


5512 


8 


12800 



The strength of manila is about \ that of hemp. 



Hand Book of Calculations. 



77 



SIZES, STRENGTH, ETC., OF ROPE. 
To find the strength of ropes. 

Rule. 

Multiply the square of the circumference by 200, the 
product will be the weight in pounds the rope will bear with 
safety. 

Example. 

What weight will a L inch rope bear in safety ? 

4x4=16 the square of the girth or circumference. 
Multiply by 200 



3200 Ans. See Table for same result. 
Table showing what weight a good hemp cable will bear in safety. 



infer- 
ence. 


Pounds. 


Circumfer- 
ence. 


rounds. 


Circumfer- 
ence . 


Pounds. 


6. 


4:320. 


9.50 


10830. 


13. 


20280 


6.50 


5070. 


10. 


12000. 


13.50 


21870 


7. 


5880. 


10.50 


13230. 


14. 


23520 i 


7.50 


6750. 


11. 


14520. 


14.50 


25230 


8. 


7680. 


11.50 


15870. 


15. 


27000 


8.50 


8670. 


12. 


17280. 


15.50 


28830 


9. 


9720. 


12.50 


18750. 




j 



To ascertain the strength of Cables. 

KULE. 

Multiply the square of the circumference, in inches, by 120 
and the product is the weight the cable will bear, in pounds, 
with safety. 

Example. 

What weight will a 12 inch cable support with safety ? 
12X12 = 144 
Multiply by 120 



2880 
144 



L7280 Ans. 



j8 Hand Book of Calctdations. 

SIZES, STRENGTH, ETC., OF ROPE. 

Note. 

Tables for strength of rope are frequently made to show the 
breaking strain, and then i to ? taken as the safety limit. In 
the two tables given the allowance is alrerdy ma'de, but for 
manila rope a further deduction should be made of %. 

Wet ropes, if small, are a little more flexible than dry; if 
large a little Ipss flexible. 

Tarred ropes are stiffer by about i, and in cold weather some- 
what more so. The stiffness of ropes increases after a little 

rest. 

The girth of a rope and its circumference are the same. 



IRON AND STEEL WIRE ROPE. 

The use of a round endless wire rope running at a great 
velocity in a grooved sheave, in place of a flat belt running on 
a flat-faced pulley, constitutes the transmissio?i of power by 
wire ropes. The distance to which this can be applied ranges 
from fifty feet up to about three miles. 

Ropes of wire — steel and iron — are made up to three inches 
in diameter, but the ordinary range in the sizes used is small, 
being from f diameter to 1^ in a range of 3 to 250 horse power. 

Two kinds of wire rope are manufactured. The most pliable 
variety contains 19 wires to the strand, and is generally used 
for hoisting and running rope: ropes with twelve wires and 
seven wires in the strand are stiff er, and are better adapted for 
standing rope, guys and rigging. 

Wire rope is as pliable as new hemp of the the same strength. 
It is manufactured either with a wire or rope center; the latter 
is more pliable than the former and will wear better where 
there is short bending. 



Hand Book of Calculations. 



19 



Table of AVike Rope. 
Hope of 133 Wires (10 wires to a strand.) 



Diam. 


Circumf. 
Ins. 


Pounds 

per foot 

run. 


Breaking load, lbs. 


Minimum diam. of drum 
in feet. 


Ins. 


Iron. 


Cast steel. 


Iron. 


Cast steel. 


H 


6* 


8.00 


148000 


310000 


8 


9 


2 


6 


6.30 


130000 


250000 


7 


8 


If 


5* 


5.25 


108000 


212000 


6.5 


7.5 


If 


5 


4.10 


88000 


172000 


5 


6 


4 


H 


3.65 


78000 


154000 


4.75 


5.5 


H 


41 


3.00 


66000 


126000 


4.5 




ii 


4 


2.50 


54000 


104000 


4 


5 


4 


34 


2.00 


40000 


84000 


3.5 


4.5 


l 


3* 


1.58 


32000 


66000 


3. 


4 


i 


2* 


1.20 


23000 


50000 


2,75 


3.75 


i 


2i 


0.88 


17280 


36000 


2.5 


3.5 


t 


2 


0.60 


10260 


28000 


2 


3 


A 


If 


0.44 


8540 


18000 


1.75 


2.75 


1 


U 


0.35 


6960 


15000 


1.5 


2 


t 


li 


0.26 


5000 




1 





Hope of 40 Wires (7 ivires to the strand,) 









Breaking load, lbs. 




Circumf. 


Pounds per foot run. 
















Iron. 


Cast steel. 


u 


4f 


3.37 


72000 


124000 


If 


*i 


•>.:: 


60000 


104000 


i* 


3f 


2.28 


50000 


88000 


i* 


3f 


1.82 


40000 


72000 


l 


3 


1.50 


32000 


60000 


i 




1.12 


•^4600 


44000 


I 


21 


0.88 


17600 


34000 


l l 
Iff 


H 


0.70 


15200 


28000 


t 


n 


0.57 


11600 


22000 


a 


ii 


0.41 


8200 


16000 


i 


if 


0.31 


5660 


12000 


1 <: 


H 


0.23 


4260 






u 


0.19 


3300 


8000 




i 


0.16 


2760 


6000 


» 
r 


i 


0.125 


2060 


.... 



Note. 
In the tables given (Jno. A. Roebling's Sons Coy. ) take I to } 
M the safe working load. 



So Hand Book of Calculations. 

SIZES, STRENGTH, ETC., OF ROPE. 

Example. 

What is the safe working load of a 2 inch cast steel wire rope. 
Now then: 

For breaking weight see Table = 28, 000 lbs. 

Divide by 7)28,000— for safety. 



4,000 lbs. Ans. 

GENERAL TABLE. 
Breaking Strain of Hope. 

3,000 lbs. per square inch of section for man] la. 

6,000 " « " hemp. 

12,000 " " " iron wire. 

24,000 " . " " steel wire. 

EULE. 

Multiply area of the rope in square inches by the figures in 
the list for kind of rope. 

Example. 

What (by the above rule) is the breaking strain of a 5 inch 
manila rope. Now then: 

5 inch rope = 1 A nearly, diam. 
The area of 1.6 = 2 inches nearly. 
Multiply by 3 COO per general rule. 



6000 lbs. = Breaking strain. 



CAUTION. 

The utmost care must be exercised in the use of any tables or 
rules for strength and safety of rope of wire or hemp and iron 
chain — a judgment of materials, amount of wear, and finish of 
manufacture, as well as the known integrity of the makers — all 
have to be taken into the calculations. 



Hand Book of Calculations. 81 



<3= YV < T1 =D Q== ^— ^T^ q= ^S$y =D 

Fig. 23. Fig. 24, Fig. 25. 

IRON CHAINS. 



Chains are constructed of round rolled iron formed with 
open links, Figs. 23 & 24, or with stud links, Figs. 25, 
26, 27 & 28. 

The cuts represent different kinds of chain, viz., Fig. 23 the 
Circular Link; Fig. 24 the Oval Link; Fig. 25 the Oval Stud 
Link, with pointed stud; Fig. 26 the Oval Link, with broad 
headed stud; Fig. 27 the obtuse-angled Stud Link, and Fig. 28 
the parallel sided Stud Link. 

The standard proportions of the links of chains, in terms of 
the diameter of the bar iron from which they are made, are as 
iollows : 

Extreme length. Extreme width. 

Stud-link, 6 Diameters, 3.6 

Close-link, 5 " 3.5 

Open-link, 6 « 3.5 

Middle-link, 5.5 " 3.5 

End-link, 6.5 " 4.5 

Example. 

What is the largest chain of the stud link pattern which can 
be made out of 1 inch iron? 

Diam. of bar=l inch. 

Multiply by 6 length of link. 

6 in. 
and for width, multiply 1 in. by 3 T % =3. 6 inches. 

Answer. — The links should be 6x3A, or less. 



82 



Hand Book of Calculations, 



IRON CHAINS. 




Fig. 26. 



d 



^^ 



V! 



u 



Fig. 27. 



Fig. 28. 



Trautwine's Table of Strength of Chains. 

Chains of superior iron will require i to % more to break 
them. 



Diam of rod 
of which 


Weight of 

chain per 

ft. run. 


Breaking strain 


Diam of rod 
of which 


Weight of 

chain per 

ft. run. 


Breaking strain of 


the links 
are made, 


of the chain. 


the links 
are made. 


the chain. 


Ins. 


Pds. 


Pds. 


Tons. 


Ins. 


Pds. 


Pds. 


Tons. 


3-16 


.5 


1731 


.773 


1 


10.7 


49280 


22.00 


X 


.8 


3069 


1.37 


1# 


12.5 


59226 


26.44 


5-16 


1. 


4794 


2.14 


IX 


16. 


73114 


32.64 


H 


1.7 


6922 


3.09 


IX 


18.3 


88301 


39.42 


7-16 


2. 


9408 


4.20 


1* 


21.7 


105280 


47.00 


l A 


2.5 


12320 


5.50 


1H 


26. 


123514 


55.14 


9-16 


3.2 


15590 


6.96 


IX 


28. 


143293 


63.97 


H 


4.3 


19219 


8.58 


lji 


32. 


164505 


73.44 


11-16 


5. 


23274 


10.39 


2 


38. 


187152 


83.55 


X 


5.8 


27687 


12.36 


2X 


54. 


224448 


100.2 


13-16 


6.7 


32301 


14.42 


2/ 2 


71. 


277088 


123.7 


H 


8. 


37632 


16.80 


2X 


88. 


335328 


149.7 


1546 


9. 


43277 


19.32 


3 


105. 


398944 


178.1 



Ton of 2240 lbs. 

The weight of close link chain is about three times the weight 
of the bar from which it is made, for equal lengths. 

Kane von Ott. — An authority comparing the weight, cost 
and strength of the three materials, hemp, iron wire and chain 
iron, concludes that the proportion between the cost of hemp 
rope, wire rope and chain is as 2: 1:3; and that therefore for 
equal resistance, wire rope is only half the cost of hemp rope, 
and a third of the cost of chains. 



Hand Book of Calculations. 83 



DECIMAL FRACTIONS. 



A decimal fraction is one whose denominator is always 10 or 
100 or 1000 or some other power as it is called of 10, but its 
numerator may be any number. For example to, rho, toW are 
all three decimal fractions. 

tV is written .1 and is in value one-tenth of a whole number. 

T V " .7 " " seven-tenths 

tU " .01 " " one-hundreth " 

rcW " .001 " " one-thousandth " " 

So it will be seen that, in decimals, by placing a figure one 
place to the right makes it a tenth of what it was before, just 
as in whole numbers. Thus: 

1000 is one thousand. 
100 is one hundred. 
10 is ten. 
1 is one. 
.1 is one-tenth. 
.01 is one-hundreth. 
.001 is one-thousandth. 

If the fraction have a numerator other than 1. Then it is 
written thus; iV is expressed .5; tVo- is expressed .27; and tWo is 
expressed .407. 

The use of the dot (.) is to separate the whole number from 
the decimal. 

The first figure after the decimal point is always tenths; the. 
second figure always hundreths; and the third figure thous- 
andths, always decreasing towards the left in a tenfold ratio. 



84 Hand Book of Calculations. 

To bring a decimal fraction to a vulgar fraction. From the 
foregoing it is plain that all we have to do is to put the given 
decimal down as a numerator; and for a denominator put down 
the figure 1, with as many cyphers after it as there are figures 
in the given decimal; then reduce it to its lowest terms. 

Examples. 

Bring .25 to a vulgar fraction, ^fr = A = i Answer. 

Bring .875 to a vulgar fraction. xWir — Mf = If = ■£. 

Bring .87500 to a vulgar fraction, tVf o%% = t oo 5 o 

Hence it will be seen by the last example that annexing a 
cypher to a decimal does not increase its value at all. You 
add as many naughts to the right as you please without affect- 
ing the value of the decimal. 

To bring a wig ar fraction to a decimal. 

Attach any number of cyphers to the numerator, and divide 
this by the denominator, being sure to have a figure for each 
naught attached. 

Examples. 

Bring \ to a decimal. 

4)100 

.25 Answer. 
Bring if to a decimal. 

4)15.0000 
16 

4)3.7500 



.9375 
Examples eok Exercise. 

1. Reduce \, i, f to decimals. 

2. i( -J, f, f and ■$ to decimals. 

3. " If, If, H, A, ye, re, Te and T V to decimals. 
Engineers sometimes find it convenient to reduce a decimal 

to a particular vulgar fraction, generally quarters, eighths, six- 
teenths or thirty-seconds. This is done thus: 

Multiply the given decimal by the denominator you wish to 
bring it to, mark off as many decimals from right to left as 
were given, and whatever number is to the left of the decimal 
point is the required numerator. 



Hand Book of Calculations. 85 

Examples. 

How many eights are there in .114 ? 
.114 

8 

.912 
Answer. None, exactly, but nearly -J. 
How many sixteenths are there in .198 ? 
.198 
16 



1188 
198 



3.138 Answer, a little over T \. 

Sometimes in reducing a vulgar fraction to a decimal frac- 
tion the quotient never comes to an end, but the same number 
keeps on repeating itself as £= 1.66666, etc., without end. 
This is called a repeating decimal, is written .16. The dot 
over the 6 represents that it is a repeater. 

A decimal fraction derives its name from the Latin word 
decern, ten, which denotes the nature of its numbers. It has 
for its denominator, a whole thing as a gallon, a pound, a 
yard, etc., which articles we suppose to be divided in tenths, 
hundredths; etc. 

ADDITION OF DECIMALS. 

Place the quantities down in such a manner that the decimal 
point of one line shall be exactly under that of every other line; 
then add up as in simple addition. 

Example. 
Thus:— Add together 36.74, 2.98046, 176.4, 31.0071 and 

.08647. 

36.74 

2.98046 

176.4 

31.0071 

.08647 



247.21403 



86 Hand Book of Calculations. 

Examples for Exercise. 

1. Add together 29.0146, 3146.05, 21.09, 6.20471 and 4.075. 

2. " " 17.14, 3.9876, 207.10104, 13.1 and 146. 

3. Find the sum of 241.01+13.98+1.90246+176.2007+14.- 
125. 

4. Find thesum of 27.27+1.125+147.5+16.0125+170.9875. 

SUBTRACTION OF DECIMALS. 

Place the lines with decimal point under decimal point, as in 
Addition. If one line has more decimal figures than another, 
put naughts under the one that is deficient till they are ^qual, 
then subtract as in simple subtraction. 

Examples. 

From 146.2004 take 98.9876. 

146.2004 

98.9876 



47.2128 Answer. 
From 4.17 take 1.984625. 

4.170000 
1.984625 



2.185375 Answer. 

Examples for Exercise. 

1. From 46.24 take 17.09864. 

2. " .2406 " .1400726. 

3. Find the value of 240.-27.7065. 

4. " " 19.72461-3.9827. 

MULTIPLICATION OF DECIMALS. 

Rule. 

Multiply as in common multiplication without taking notice 
of the decimal point, add up and so get the product; then 
count all the figures after or at the right of the decimal points 
in the multiplier and multiplicand; count from the right 
towards the left of the product as many figures as the sum of 
the decimals just counted; put a decimal point before the fig- 
ures and you have the answer. 



Hand Book of Calculations. 8j 



Example. 

Multiply 27.62 by 5.713. 

27.62 
5.713 



8286 
2762 
19334 
13810 

157.79306 

The product first stood 15779306, but as there are altogether 
five decimal figures in the question, we count five beginning at 
the last or figure 6, and place a decimal point before the figure 
that stands in the fifth place. The answer is 157.79306. 

Example. 

Multiply .00072 by 0.502. 

.00072 
•0502 



144 

3600 

36144 



The product is 36144, but as we have nine places of decimals 
in the example, we must have the same number of decimals in 
the product. This is done by putting cyphers to the left of 
the product. The answer is .000036144. 

Examples for Exercise. 



1. 


Multiply 


724.02 by 23.14. 


2. 


a 


23.567 by 3.25. 


3. 


t< 


.3024 by .3055. 


4. 


a 


.5052 by .0025. 


5. 


a 


.0002 by .00101. 


6. 


a 


176401 by 76.43. 



88 Hand Book of Calculations. 



DIVISION OF DECIMALS. 

1. When the divisor is a whole number: divide as in simple 
division, only when you come to the decimal point place a 
point under it in the quotient. 



Divide 763.5676 by 4. 
Divide 1537.27 by 8. 



Examples. 

4)763.5676 
190.8919 

8)1537.27 



192.15875 
After saying 8 into 47 goes 5 times and 7 over, make this 7 
into 70; 8 into 70 goes 8 times and 6 ov T er; 8 into 60 goes 7 
times and 4 over; 8 into 40 goes 5 times. 
Divide 72.6432 by 24. 

24 is 6 times 4. Divide by 6, and then the quotient by 4. 
( 6)72.6432 

24-^ 

( 4)12.1072 



3.0268 Answer. 
Divide 7196.148 by 1728. 

1728)7196.148(4.1644 etc. or 

6912 f 12)7196.148 



2841 1728 -{ 12)599.679 

1728 I 

t 12)49.7325 



11134 

10368 4.1644375 

7668 
6912 

7560 
6912 



648 



Hand Book of Caladations. 8? 

DIVISION OF DECIMALS. 

Here after the 8 is brought down it goes 4 times, and the 
remainder is 756; to this attach an 0, and let it go again, and 
so on as far as it is thought necessarj^. 

When the number of decimal figures in the divisor is less than 
that in the dividend, divide without taking notice of the deci- 
mals; tnen subtract the number- of decimals in the divisor from 
the number in the dividend; the remainder will be the number 
to mark off in the quotient. 

Examples. 

Divide 172.4025 by .5. 

.5)172.4025 



34.4805 
Here we say 1 from 4 leaves 3 : then mark off 3 decimals in 
answer. 
Divide .0041275 by .25. 

5).0041275 
25 



5) 8255 

1651 
Here it is 2 from 7 leaves 5 : mark off 5 in the quotient; we 
caunot because there are only 4; then attach a cypher to the 
left and it becomes .01651 Answer. 
172.4025 by .5. 

First shift the decimal back one place and it becomes 1724.- 
025 by .5. 

Then 5)1724.025 



3448.05 which is the same as before. 
Divide .0041275 by .25. 

First shift the decimal back two places and it becomes 
00.41275. 

5). 41275 



25 



5).08255 



,01651 (1651) which is the same as before. 



<go Hand Book of Calculations. 

4. When the decimals in the divisor are more than those 
in the dividend. First equalize the decimals by attaching 
naughts to that which has the least; then leave out the points 
altogether and divide as in simple division, and the quotient is 
whole numbers. If there is a remainder after this, attach a 
naught to it and again divide; this will give the first decimal 
of the quotient; to the remainder again attach a naught; again 
divide for the second decimal figure and so on as far as may be 
thought necessary. 

Examples. 
Divide 1.1 by .275. 

Equalize the decimal figures thus: 1.100 by .275, and then 
leave out the decimal point and divide thus: 
275)1100(4 
1100 

Answer 4 whole numbers. 

1562.5 by .00025.* 

Equalize 1562.50000 by .00025, then leaving out the points 
divide as in Simple Division. 



25 



C 5)156250000 
( 5) 31250000 



6250000 Answer. 

Divide 147.24 by .84625. 

Equalize 147.2400 by .84625. 

84625)1472400(173.99, etc. Here it goes once, then seven 

84625 times and then 3 times; and as 

~~~~~~ there are no more figures left to 

626150 , . , ,, ■ ■ . ° , , 

bring down these 173 are whole 

5 97325 numbers. To find the decimals 

337750 attach a cypher to the remain- 

.oq*k der 83 875, and it goes 9 times; 

this is put m the quotient as .9; 

838750 to the remainder 77125 attach 

761625 another cypher and it goes 9 

again; put this 9 after the for- 

771250 mer one; attach another cypher 

761625 to the remainder if necessary 

and continue as far as you please. 



Hand Boole of Calculations. gi 

Examples for Exercise. 
1. Divide 713.915 by 5. 



2. 


a 


39.5424 by 8. 


3. 


a 


.936571 by 12. 


4. 


a 


23(30.715 by 15. 


5. 


a 


87916.05 by 88. 


6. 


tc 


375.4329 by 80. 


7. 


a 


17624 by .6725. 


8. 


a 


46.59005 by 7.25. 


9. 


a 


210.75 bv 24.25. 



UNITED STATES MONEY. 

United States money is added, substracted and divided in all 
respects like Decimal Fractions. 

50 cts. — $i. 12| cts. = $}. 

25 " =$ii 8i " =$ T V 

20 " =8i 6i " ==$iV. 

16f " =$£. 5 " = $ ¥ V 

The dollar is the «»iY with the decimal point placed after 
it. Cents occupy two places, hence if the number to be 
expressed is less than 10 a cypher must be prefixed to the figure 
denoting them. Mills occupy the place of thousandths. 
Example. Two dollars and eight cents is written $2.08. 



PERMUTATION. 



Permutation is the method for ascertaining how many differ- 
ent ways any given number of persons or things may be varied 
in their positions. 

Rule. 

Multiply all the terms of the natural series continually to- 
gether, and the last product will be the number of changes 
which may be effected. 

Example. 
How many positions in a row can 8 things be placed. 
12345678 
And 1x2x3x4x5x6x7x8 = 40,320 Ans. 



9 2 



Hand Book of Calculations. 



MENSURATION. 



Mensuration is the art of measuring things which occupy 
space; the art is partly mechanical, and partly mathematical, 
hence can be illustrated with drawings to aid in the better 
understanding of the arithmetical problems connected with the 
art. 

There are three kinds of quantity in space, viz., length, sur- 
face and solidity; and there are three distinct modes of meas- 
urement, viz., mechanical measurement, geometrical construc- 
tion, and algebraical calculations. The last two modes can 
only be done by calculations, but in mechanical measurements, 
they are made by the direct application of rules, tape-lines and 
chains. 

Lengths are measured on lines, and the measure of a length 
of a line is the ratio which the line bears to a recognized unit 
of length, the inch, foot, or anile, determined by reference to 
brass rods kept by the U. S. government at Washington as a 
standard. The use of the " rules " is called direct measurement. 

The second kind of quantity to be measured is surface. This 
sort of measurement is never done, directly or mechanically 
but always by the measurement of lines, as will be seen both 
under this division and under the sections relating to geometry. 

The third species of quantity is solidity. Direct measure- 
ments of solid quantities, consists simply in filling a vessel of 
known capacity, like a bushel or gallon measure, until all is 
measured. The geometrical mode of computing solids is the 
one hereafter shown by examples and illustrations. 



Hand Book of Calculations. 



93 



MENSURATION. 



To find the length of the curved line, called the circle; that is, 
to find the circumference of a circle. 




Fig. 29. 
Rule. — Multiply 3.1416 by the diameter. 

Example. 
"What is the circumference of a circle whose diameter is 3 
inches ? 

3.1416 
3 



Answer, 9.4248 inches. 
Example. 
What is the circumference of a circle whose diameter is 4£ 
inches ? 





Diamet 

a 
a 
t< 
a 


3.1416 
4.5 






157080 
125664 




1. 

2. 

3. 
4. 
5. 

6. 


Answer, 14.13720 inches. 

Examples por Exercise. 

er 5 in., required the circumference? Ans. 
5.6 in. " 
2.5 in. " 

4 in. " " " 
3iin. « 
7fin. " 


15.708 
17.59296 

7.854 
12.5664 
10.2102 
23.9547 



94 Hand Book of Calculations. 

MENSURATION. 
To find the area of a circle. 




Fig. 30. 
Kule. — Multiply .7854 by the square of the diameter. 

Example. 
The diameter of a circle is 3 inches, find its area. 
3 .7854 

3 9 



9 Answer, 7.0686 square inches. 

Example. 
The diameter of a circle is 3.5 inches, find the area. 
3.5 . .7854 

3.5 12.25 



175 39270 

105 15708 

15708 

12.25 7b54 



Answer, 9.621150 square inches. 
A very easy method of multiplying by .7854 is shown by the 
following: — 

12.25 

.7 . ' 

8575 

8575 

17150=product of first line of 
17150 multiplication X 2. 



9.621150 
The method of procedure is as follows : — The number is 



Hand Book of Calculations. 95 

MENSURATION. 

multiplied first by 7 by common multiplication, this line is put 

down a second time, only removed one place to the right instead 

of to the left, as in ordinary multiplication. This line is now 

multiplied by 2, and its result put down one place to the right, 

this line is again put down one place to the right and the 

sum of these products is the same as if we had multiplied by 

.7854 in the ordinary manner. The process may be rendered 

clearer if the reason for the method is explained. If we put 

down the number 7, then one place to the right put it down 

7 again; then multiply it by 2 and put product one 

7 place to the right, then put this down again one place 

14 to the right, and add them all up, we clearly get the 

14 number 7854, therefore if we multiply in this order 

we get the same result as if .7854 had been used full 

7854 out. 

Examples. 

371281GX.7854 

7 



25989712 
25989712 
51979424 
51979424 

2910045.6864 

46219872 X. 7854 

7 



323539104 
323539104 

647078208 
647078208 



36301087.4688 
Examples for Exercise. 

1. The diameter of a circle is 5 inches, find its area. 

Answer, 19.635 square inches. 

2. " of a circle is 4.6 inches, find its area. 

Answer, 16.619064 square inches. 

3. u of a circle is 6 J inches, find the area. 

Answer, 37.122421875 square inches. 



<p6 Hand Book of Calculations. 

MENSURATION. 
To find the Circumference of an Ellipse. 




Multiply 3.1416 by half the sum of the two diameters; the 
product will be the circumference nearly. 

Example. 

What is the circumference of an ellipse whose diameters are 
9 and 7 feet? 

9 3.1416 

7 8 



2)16 25.1328 feet, Answer. 

8 

Example. 

What is the circumference of an ellipse whose diameters are 
5J and 4£ respectively? 

5.75 3.1416 

4.25 5 



2)10.00 15.7080 



Hand Book of Calculations. 



97 



MENSURATION. 
To find the Area of an Ellipse. 




Fig. 32. 

Rule. 
Multiply . 7854 by the product of the diameters. 

Example. 

What is the area of an ellipse whose diameter is 5 J and 4-JP 
5.75 24.4375 

4.25 .7854 



2875 
1150 
2300 

24.4375 



977500 
1221875 
1955000 
1710625 

19.19321250 



Example. 

What is the area of an ellipse whose diameters are 7 and 9 
feet? 

7 .7854 

9 63 



63 



23562 
46124 



40.4802 square feet, Answer. 



9 s 



Hand Book of Calculations. 



MENSURATION. 

To find the area of a Square. 

Note. 
A Square is a figure having all its angles right 
all its sides equal. 



?, and 




Fig. 33. 
Eule. 
Multiply the base by the height; that is multiply the length 
by the breadth. 

EX A3IPLE. 

What is the area of a square whose side is 2-J feet? 

2.5 
2.5 



125 
50 



Answer, 6.25 square feet. 
To find the area of an Oblong. 

Note. 
An Oblong is a figure whose angles are all righi tingles, tut 
whose sides are not all equal, only the opposite sides are equal. 




Fig. 34 



Hand Book of Calculations. pp 

MENSURATION. 

Rule. 
Multiply the length by the breadth. 

Example. 



What is the area of a rectangular figure whose base is 12 feet 

12 



and height 8 feet? 



Answer, 96 square feet. 
To find the area of a Parallelogram. 

Note. 

A Parallelogram is a figure ivhose opposite side are parallel; 
the square and oblong are parallelograms; so also are other four- 
sided figures, tvhose angles are not right angles. It is these 
latter whose area we now want to find. 

Rule. 
Multiply the base by the perpendicular height. 

Example. 

Find the area of a parallelogram whose base is 7 feet and 
height 5i feet? 




Fig. 35. 

5.25 

7 

Answer, 36.75 square feet. 



100 



Hand Book of Calculations. 




MENSURATION. 

To find the area of a Triangle. 

Note. 

A Triangle is a figure bounded by three sides, and is half a 
parallelogram; hence the 

Rule. 

Multiply the base by half the perpendicular height. 



Fig. 36. 
Example. 

The base of the triangle is 12 feet, and it is also 12 feet high, 
what is its area? 

Half the height ~6 feet; and 12x6 = 72 square feet area. 
To find the area of a Trapezium. 

Note. 

A Trapezium is any four -sided figure that is neither a recU 
angle, like a square or oblong, nor a parallelogram. 

Rule. 

Join two of its opposite angles, and thus divide it into two 
triangles. 

Measure this line, and call it the base of each triangle. 

Measure the perpendicular height of each triangle aboye the 
base line. 



Hand Book of Calculations, 



10 r 



MENSURATION. 



Then find the area of each triangle by the last rule; their 
sum is the area of the whole figure. 




Fig. 37. 
To find the area of a Trapezoid. 

Note. 
A Trapezoid is a trapezium having two of its sides parallel. 

Rule. 
Multiply half the sum of the two parallel sides by the per- 
pendicular distance between them. 

k ? >i 




Fig. 38. 
Let the figure be the trapezoid, the sides 7 and 5 being 
parallel; and 3 the perpendicular distance between them. 

Example. 

Find the area of the above trapezoid, the parallels being 7 
feet and 5 feet, and the perpendicular height being 3 feet. 

7 
5 



2)12 

6 And 6x3 = 18 square feet. 



102 Hand^ Book of Calculations. 

MENSURATION. 
To find the Surface or Envelope of a Cylinder. 

Eule. 

Multiply 3.1416 by the diameter, to find the circumference; 
and then by the height. 

Example. 

What is the surface of a cylinder whose diameter is 9 inches, 
and height 15 inches. 

3.1416 
9 



28. 2 7 44= circumference. 
15 



14l37-<0 

282744 



424.1160 area of surface in square inches. 
To find the Surface or Envelope of a Sphere. 

Note. 

The surface of a sphere is equal to the convex surface of the 
circumscribing cylinder; hence the 

Eule. 

Multiply 3.1416 by the diameter of the sphere, and: this again 
by the diameter; because in this case the diameter is the height 
of the cylinder; 

Or multiply 3.1416 by the square of the diameter, of the 
sphere. 

Example. 

What is the surface of a sphere whose diameter is 3 feet? 
3.1416 

9=3 2 



28.2744 area cf surface in square feet. 



Hand Book of Calculations. 



ioj 



CONTENTS OF SOLIDS. 

To find the Contents of a Rectangular Solid, 

Rule. 
Multiply the length, breadth, and height together. 




"What is the content of a rectangular solid whose length is 5 
feet, breadth 4 feet, and height 3 feet ? 

5 feet 
4 feet 

20 square feet of base 
3 feet 

60 cubic feet 



104 



Hand Book of Calculations, 



CONTENTS OF SOLIDS. 



To find the cubic cantents 
of a Solid Cylinder. 

Rule. 

Find the area of the base, 
and multiply this by the 
height or length. 




Fig. 40. 



Example. 



"What are the cubic contents of a cylinder whose diameter is 
4 feet, and height or length 7£ feet? 



4 
4 

16 



.7854 
16 

47124 

7854 



12.5664=area of base in square feet 
7.5=height or length in feet 

628320 
879648 



Answer, 94. ; 4b00 cubic feet. 



Hand Book of Calculations. 



'°5 



To find the Cubic contents of a Sphere. 

Rule. 
Multiply .7854 by the cube of the diameter, and then take § 

of the product. 

Example. 

Find the cubic contents of a sphere whose diameter is 5 feet* 



5 
5 


.7854 
125 


25 
5 

25 = 5 3 


39270 
15708 

7854 


98.1750 

2 




3)196.3500 



Answer, 65. 4500 cubic feet. 
To find the cubic contents of a Frustrum of a Cone. 
[A frustrum of a cone is the lower portion' of a cone left after 
the top piece is cut away. 



Rule. 



Find the sum of the squares of the 
two diameters (d, D), add on to this 
the product of the two diameters 
multiplied by .7854, and by one- 
third the height ("h.") 



Example. 

Find the cubic contents of a safety valve weight of the fol- 
lowing dimensions: — 12" large diameter, 6" small diameter, 4" 
thick. Now: 

144+36+72 X. 7854x1.33 
252x.7854xl.33x = ^63.23 Ac. cubic inches. 




lo6 Hand Book of Calculations. 



VULGAR FRACTIONS. 



A fraction means a part of anything. If an apple be cut 
into eight equal parts each part will be called an eighth of the 
whole apple, and is written -g-. This eighth is a fraction. If 
we had 3 or 5 or 7 of these pieces of the apple, we would rep- 
resent it by §, £> or $, as the case might be. All these are 
fractions. 

A yulgar fraction is always represented by two numbers (at 
least), one over the other and separated by a small horizontal 
line. The one above the line is always called the Numerator, 
and the one below the line the Denominator. 

The denominator tells us how many parts the whole thing 
has been divided into, and the numerator tells us how many of 
those parts we have. Thus in the fraction f above, the eight 
is the denominator, and shows that the apple has been divided 
into eight equal parts; and three is the numerator, and shows 
that we have three of those pieces or parts of the apple. 

A proper fraction is one whose numerator is less than the 
•denominator, as f or f. 

An improper fraction is one whose numerator is more than 
its denominator as f or f . 

J means more than a whole one, because f must be a whole 
one. Thus f will be 3 thirds -f- 3 thirds -j- 2 thirds or 2f, and 
this form of fraction is called a mixed number. 



Hand Book of Calculations. ioj 

VULGAR FRACTIONS. 

1 . To reduce an improper fraction to a mixed number. Divide 
the numerator by the denominator; the quotient is the whole 
number part, and the remainder is the numerator of the frac- 
tional part. 

Example: V=2f. Example: ¥=5. Example: ¥=3f. 

2. To reduce a mixed number to an improper fraction. 
Multiply the whole number part by the denominator, and add 
on the numerator; the result is the numerator of the improper 
fraction. 

Example: 2f=V-. Example: 5£=V> Example: 3f=V L . 

3. To reduce a fraction to its loivest terms. Divide both 
numerator and denominator by the same number; if by so 
doing, there is no remainder. 

Example. 
Reduce &. Here 4 will divide both top and bottom without 
a remainder. Divide by 4. 

4)A-|. 

The meaning of this is, that if you divide a thing into 12 
equal parts, and take 8 of them, you will have the same as if 
the thing had been divided into 3 equal parts and you had 2 of 
them. 

Example. 

Reduce tVsVt to its lowest terms. First divide top and bot- 
tom by 12 and it becomes tW*; then divide top and bottom 
again by 12 and it becomes H; 12 will again divide them and 
it becomes ^, which is its lowest term. 

Examples for Exercise. 
Reduce to their lowest terms &; H; H; tVA; iff and If Si 

4. To reverse the last rule. To bring a fraction of any de- 
nominator to a fraction having a greater denominator. 

See how often the less will go into the greater denominator 
and multiply both numerator and denominator by it. The 
result is the required fraction. 



108 Hand Book of Calculations. 

Example. 
Bring \ to a fraction whose denominator is 8. 
Here 2 goes in 8, 4 times; then multiply the numerator and 
denominator of i by 4=|, which is the required fraction. 

Example. 

Bring f to a fraction whose denominator is 15. 

Here 3 goes into 15 five times; then f becomes H. 

5. If you have a fraction of a fraction, as -J of \, it is called 
a compound fraction, and should always be reduced to a simple 
fraction, by multiplying all the numerators together for a new 
numerator, and all the denominators together for a new denom- 
inator; then, if necessary, reduce this fraction to its lowest 
terms. 

Example. 

f of f of f. Eeduce this to a single fraction: 3x2x4=24; 
and 4x3x9=108. 

Thus tVf is the fraction. Eeduce this 12 ) t %f=I. 



CANCELLATION. 



This is a method of shortening problems by rejecting equal 
factors from the divisor and dividend. 

The sign of cancellation is an oblique mark drawn across 
the face of a figure as fi, 0, t- 

Cancellation means to leave out; if there are the same num- 
bers in the numerator and the denominator they are to be left 
out. 

Example. 

f of f of f. Here the 3 in the first numerator and the 3 in 
the 2d denominator are left out; also 4 of the first denomina- 
tor and the last numerator, thus: 

Ans.*x!-X^ 



There is another way of cancellation. 



Hand Book of Calculations. log 

Example. — 3 of J of it of T 9 A=by cancellation, thus: 

gofjqfgof*. = 7 =± 

$ W in 3X2X34 204 
3 A 34 

2 
The process is as follows: — The first numerator 2 will go into 
8 the denominator of the second fraction 4 times; the denomi- 
nator of the third fraction 18 will go into 90, the numerator of 
the last quantity 5 times. The numerator of the second frac- 
tion 3, will go into the denominator of the first fraction, 3 times; 
5 will go into 170, 34 times; 2 will go into 4 twice, and 2 into 
14, 7 times, and as we cannot find any more figures that can be 
divided without leaving a remainder we are at the end, and the 
quantities left must be collected into one expression. On 
examination we have 7 left on the top row, this is put down at 
the end as the final numerator; on the bottom we have 3, 2, 
and 34, these multiplied together give us 204, which is the 
final denominator. 

Rules for Cancelling. 
1. Any numerator can be divided into any denominator pro- 
vided no remainder is left, and vice versa, thus: 



5 15 



*_of*J = ! 

$ 1$ 2 



2 

2. Any numerator and denominator may be divided by the 
same number, provided no remainder is left, and the decreased 
value of such numerator and denominator be inserted in the 
place of those cancelled, — 

5 Here 8 is divided by 4, and 20 can also be 

$ of J20 divided by the same number without leaving 

^ 31 an y remainder. Answer *V. 



;l 



Example. 

n n 17 3x2xi7 102 

3 t 

2 



i io Hand Book of Calculations. 

Examples for Exercise. 

Eeduce to their simplest form the fractions: 
1. I off off of *. 

This can be done by cancelling. 



I of * of ? of \ = I Answer. 



2. | of if of i 
By cancelling. 



* of ■!* of £ = JL = 1 = I Answer. 

10 4x3 12 4 

4 3 

3. f of J of if. 
By cancelling. 

% 

l of 1 of % = JL = 4 Answer. 

Jfr >< 10 2x3 6 

2 3 

ADDITION OF FRACTIONS. 

Add together -J, f and J-. Here it is evident that the sum 
will be f or 1^. Hence the rule: Bring all the fractions to the 
same common denominator, add their numerators together for 
the new numerator, and reduce the resulting fraction to its 
simplest form. 

Examples. 

What is the sum of i+i=i+l=J Ans. 

What is the sum of f +|+t +l=¥=l|. 

To bring fractions having different denominators to fractions 
having one common denominator. 

Rule. 

1. Put all the denominators down in a row; cancel all that 
are alike except one; also cancel any that will divide into 
another one without remainder. 

2. If there is any number that will divide two or more of 
those left, then divide by it, putting down those numbers also 
that cannot be divided. Repeat this till all the numbers are 
prime numbers. 



Hand Book of Calculations. 1 1 1 



3. Then multiply all these prime numbers together, and 
their product by all the divisors: the result will be the com- 
mon denominator for all the fractions. 

4. Lastly, divide this common denominator by the denomi- 
nator of the first fraction, and multiply its quotient by its nu- 
merator; the product is the new numerator required. Repeat 
this for each fraction. 

Example. 

\y I, v, I, h h tzj t 5 6, £ and f. Bring these fractions to 
others having a common denominator. 

•> 6 3 8 3 8 12 16 8 4 

There are 2 figures 3, cancel one of them, there are 3 figures 

8, cancel 2 of them; next the 2, 8 and 4 will each go into 16, 

therefore they must be cancelled; the 6 and 3 also, because 

they will each divide into 12; then there only remain the 12 

and 16, place them as below and divide them by 4. See Article 

2 of rule 

4)12 16 

3 4 

Then multiply the 3 by the 4 = 12, and this 12 by the 
divisor 4 = 48, the common denominator. 

Lastly, bring each fraction to one having the denominator 
48 by rule (article 4) heretofore given. 

HU'liAAU 
Ans. « V* i I it « U H if A ft 
Example. — Add together |, |, I, T V and $■ 
2) , 3, 4, 10, 

3,2, 5 

2 

10 
3 

30 
2 Divisor 



60 Common denominator 



48+40+45+42+30 
60 • 



112 Hand Book of Calciilations. 

Examples for Exercise. 

1. Add together -J, f, \, and f . 

2. " « hi, f, andf. 

3. " " #, to, t 5 * and |. 

4. " " i, i i, h h it and-^. 

5. « « |, I, i, I, A and f . 

SUBTRACTION OF FRACTIONS. 

Bring the fractions to others having a common denominator, 
as in addition, and subtract their numerators. 

Examples. 
From -J- subtract f = f =•■£. 
From -J take i. -^ = I = i. 

7 _8. 7-6 l 

T6 £ T6 " lb ' 

What is the difference between ■£ of f and J of 1-J ? 

i of f = |; and i of H = i of | = |. 
Therefore it is f — § = 0. 
Which is the greater, J of T 9 o or f of A ? 

■J of T 9 d = to ; and f of tit =1. 
Therefore it is tV and |. 

27 and 40 



90 
Then f of A is the greater by $$. 

Examples for Exercise. 



l. l-i; t-i; t-A; t-i. 

2 5 3.3 2. 7 1 

. T 3 ? ^ 7. X6 7. 

3. What is the difference between f of f and li of I ? 

4. Which is the greatest, 3f of 2f or 8J- of 1| ? 



MULTIPLICATION OF FRACTIONS. 

First bring each fraction to its simplest form; then multiply 
the numerators together for the new numerator, and the 
denominators together for the new denominator. Reduce the 
fraction to its simplest form. 



Hand Book of Calculations. iij 







Examples 








1. 


Multiply 


1 X 1 t\ ; that 


is |X 


'. f 5 TT% fi 1> 


or 


by 


canceling 
















1 


3 












*X 


21 


3 










t 


H ' 


4 










1 


4 








The A 


: cancels into the 16 four times, and the 7 into 


the 21 


three 


times. 


Hius 1X3 = \ 


3, and 


1X4 = 4. Answer f . 




2. 


2tV of 3| 


X6iof *. 

3 5 

10 jT 

2 1 
5 

10 7 
2 X $ : 

1 


7 
x ^ 

x T 
1 

35 

~~ ~2 = 


1 

u 

3 
= 17-J- Answer, 







Examples for Exercise. 

1. Multiply ixi; fxt; AxA. 

2. " lfxlt; 5|x3A; 4fx2 T V. 

3. " }off offxf of | off. 

DIVISION OF FRACTIONS. 

Reverse the divisor and proceed as in multiplication. 
The object of inverting the divisor is convenience in multi- 
ply i: 

After inverting the divisor, cancel the common factors. 

Examples. 

J-r-li, that is, f-s-i, reverse the f and it becomes |; then the 
question is fxf = U =J Ans. 

4? of tt-irSi of 3i, that is V of »-*■¥ of ¥; cancelling 
reduces the dividend to J and the divisor to Y and we have |-5- 
V, that is H-iV= s ^ =*J Ans. 

Examples for Exercise 

1. *-H; »-i; *+i; ^H. 

2. 3±-H 5S-r-2i; ^-2f. 



U4 



Hand Book of Calculations. 



TABLE 

CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES 



Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 


0.0 






3.0 


7.0686 


9.4248 


.1 


.007854 


.31416 


.1 


7.5477 


9.7389 


«2 


.031416 


.62832 


.2 


8.0425 


10.0531 


£ 


.070686 


.94248 


':s 


8 5530 


10.3673 


A 


.12566 


1.2566 


A 


9.0792 


10.6814 


.5 


.19735 


1.5708 


.5 


9.6211 


10.9956 


.6 


.28274 


1.8850 


.6 


10.1788 


11.3097 


.7 


.38485 


2.1991 


.7 


10.7521 


11.6239 


.8 


.50266 


2.5133 


.8 


11.3411 


11.9381 


.9 


.63617 


2.8274 


.9 


11.9456 


12.2522 


1.0 


.7854 


3.1416 


4.0 


12.5664 


12.5664 


.1 


.9503 


3.4558 


.1 


13.2025 


12.8805 


.2 


1.1310 


3.7699 


.2 


13.8544 


13.1947 


.3 


1.3273 


4.0841 


.3 


14.5220 


13.5088 


.4 


1.5394 


4.3982 


.4 


15.2053 


13.8230 


.5 


1.7671 


4.7124 


.5 


15.9043 


14.1372 


.6 


2.0106 


5.0265 


.6 


16.6190 


14.4513 


.7 


2.2698 


5.3407 


.7 


17.3494 


14.7655 


.8 


2.5447 


5.6549 


.8 


18.0956 


15.0796 


.9 


2.8353 


5.9690 


.9 


18.8574 


15.3938 


2.0 


3.1416 


6.2832 


5.0 


19.6350 


15.7080' 


.1 


3.4636 


6.5973 


.1 


20.4282 


16.0221 


.2 


3.8013 


6.9115 


.2 


21.2372 


16.3363 


.3 


4.1548 


7.2257 


.3 


22.0618 


16.6504 


A 


4.5239 


7.5398 


.4 


22.9022 


16.9646 


.5 


4.9087 


7.8540 


.5 


23.7583 


17.2788 


.6 


5.3093 


8.1681 


.6 


24.6301 


17.5929 


1.7 


5.7256 


8.4823 


.7 


25.5176 


17.9071 


.8 


6.1575 


8.7965 


.8 


26.4208 


18.2212 


.9 

4** 


6.6052 


9.1106 


.9 


27.3397 


18.5354 



Hand Book of Calculations. 



JI 5 



TABLE— {Continued.) 

CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES. 



Diam. 


Area. 


Circuin. 


1 Diam. 


1 Area. 


Circum. 


6.0 


28.3743 


18.8496 


10.0 


78.5398 


31.4159 


.1 


29.2247 


19.1637 


.1 


80.1185 


31.7301 


.2 


30.1907 


19.4779 


.2 


81.7128 


32.0442 


.3 


31.1725 


19.7920 


.*3 


83.3229 


32.3584 


.4 


33.1699 


20.1062 


.4 


84.9487 


32.6726 


.5 


33.1831 


20.4204 


..5 


86.5901 


32.9867 


.6 


34.2119 


20.7345 


.6 


88.2473 


33.3009 


.7 


35.2565 


21.0487 


.7 


89.9202 


33.6150 


.8 


36.3168 


21.3628 


.8 


91.6088 


33.9292 


.9 


37.3928 


21.6770 


.9 


93.3132 


34.2434 


7.0 


38.4845 


21.9911 


11.0 


95.0332 


34.5575 


.1 


39.5919 


22.3053 


,1 


96.7689 


34.8717 


.2 


40.7150 


22.6195 


.2 


98.5203 


35.1858 


.3 


41.8539 


22.9336 


'.S 


100.2875 


35.5000 


.4 


43.0084 


23.2478 


A 


102.0703 


35.8142 


.5 


44.1786 


23.5619 


.5 


103.8689 


36.1283 


.6 


45.3616 


23.8761 


.6 


105.6832 


36.4425 


.7 


46.5663 


24.1903 


.7 


107.5132 


36.7566 


.8 


47.7836 


24.5044 


.8 


109.3588 


37.0708 


.9 


49.0167 


24.8186 


.9 


111.2202 


37.3850 


8.0 


50.2655 


25.1327 


12.0 


113.0973 


37.6991 


.1 


51.5300 


25.4469 


.1 


114.9901 


38.0133 


.2 


52.8102 


25.7611 


.2 


116.8987 


38.3274 


.3 


54.1061 


26.0752 


.3 


118.8229 


38.6416 


.4 


55.4177 


26.3894 


.4 


120.7628 


38.9557 


.8 


56.7450 


26.7035 


.5 


122.7185 


39.2699 


.6 


58.0880 


27.0177 


.6 


124.6898 


39.5841 


.7 


59.4468 


27.3319 


.7 


126.6769 


39.8982 


.8 


60.8212 


27.6460 


.8 


128.6796 


40.2124 


.9 


62.2114 


27.9602 


.9 


130.6981 


40.5265 


9.0 


0:5.0173 


28.274:5 


13.0 


132.7323 


40.8407 


.1 


65.0388 


28.5885 


.1 


134.7822 


41.1549 


.2 


66.4761 


28.9027 


.2 


136.847s 


41.4690 


.3 


07.9291 


29. 21 OS 


.3 


138.9291 


41.7832 


.4 


69.3978 


29.5310 


.4 


141.0261 


42.0973 


.5 


70.8822 


29.8451 


.5 1 


143.1388 


42.4115 


.8 


72.3823 


30.1593 


.6 


145.2672 


42.7257 


.7 


73.8981 


30.4734 


.7 


147.4114 


43.0398 


.8 


75.4296 


30.7870 


.8 


149.5712 


43.3540 


.9 


76.9709 


31.1018 


.9 


151.7468 


43.6681 



n6 



Hand Book of Calculations. 



TABLE— {Continued. ) 

CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OP CIRCLES. 



Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 


14.0 
.1 

.2 . 
.3 
.4 


153.9380 
156.1450 
158.3677 
160.6061 
162.8602 


43.9823 
44.2965 
44.6106 
44.9248 
45.2389 


8.0 
.1 
.2 
.3 
.4 


254.4690 
257.3043 
280.1553 
263.0220 
265.9044 


56.5486 
56.8628 
57.1770 
57.4911 
57.8053 


.5 
.6 

.7 
.8 
.9 


165.1300 
167.4155 
169.7167 
172.0336 
174.3662 


45.5531 
45.8673 
46.1814 
46.4956 
46.8097 


.5 
.6 

.7 
.8 
.9 


268.8025 
271.7164 
274.6459 
277.5911 

280.5521 


58.1195 
58.4336 
58.7478 
59.0619 
59.3761 


15.0 
.1 
.2 

.3 

.4 


176.7146 
179.0786 
181.4584 
183.8539 
186.2650 


47.1239 

47.4380 
47.7522 
48.0664 
48.3805 


19.0 
.1 

.2 
.3 
.4 


283.5287 
286.5211 
289.5292 
292.5530 
295.5925 


59.6903 
60.0044 
60.3186 
60.6327 
60.9469 


.5 
.6 

.7 
.8 
.9 


188.6919 
191.1345 
193.5928 
196.0668 
198.5565 


48.6947 
49.0088 
. 49.3230 
49.6372 
49.9513 


.5 
.6 

.7 
.8 
.9 


298.6477 
301.7186 
304.8052 
307.9075 
311.0255 


61.2611 
61.5752 
61.8894 
62.2035 
62.5177 


16.0 
.1 
.2 
'.S 

.4 


201.0619 
203.5831 
206.1199 
208.6724 
211.2407 


50.2655 
50.5796 
50.8938 
51.2080 
51.5221 


20.0 
.1 
.2 
!3 

.4 


314.1593 
317.3087 
320.4739 
323.6547 
326.8513 


62.8319 
63.1460 
63.4602 
63.7743 

64.0885 


.5 
.6 

.7 
.8 
.9 


213.8246 
216.4243 
219.0397 
221.6708 
224.3176 


51.8363 
52.1504 
52.4646 

52.7788 
53.0929 


.5 
.6 
. i 
.8 
.9 


330.0636 
333.2916 
3o6.5353 
339.7947 
343,0698 


64.4026 
64.7168 
65.0310 
65.3451 
65.6593 


17.0 
.1 

.2 
.3 
.4 


226.9801 
229.6583 
232 3522 

235.0618 

237.7871 


53.4071 
53.7212 
54.0354 
54.3496 
54.6637 


21.0 
.1 
.2 
'.'S 
A 


346.3606 
349.6671 
352,9894 
356.3273 . 
359.6809 


65.9734 
66.2876 
66.6018 
66.9159 
67.2301 


.5 
.6 

.7 
.8 
.9 


240.5282 
243.2849 
246 0574 
248.8456 
251.6494 


54.9779 
55.^920 
55.6062 
55.9203 
56.2345 


.5 
.6 

.7 
.8 
,9 


363.0503 
366.4354 
369.8361 
373.2526 

376.6848 


67.5442 
67.8584 
68.1726 
68.4867 
68.8009 



Hand Book of Calculations. 



n 7 



TABLE— (Continued.) 

CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES, 



Diam. 


Area. 


Clrcum. 


Diam. 


Area. 


Circum. 


22.0 


380.1827 


69.1150 


26.0 


530.9292 


81.6814 


.1 


383.5963 


69.4292 


.1 


535.0211 


81.9956 


.2 


387.07-6 


69.7434 


.2 


539.1287 


82.3097 


.3 


390.57< 17 


70.0575 


.3 


543.2521 


82.6239 


.4 


394.0814 


70.3717 


.4 


547.3911 


82.9380 


.5 


397.6078 


- 70.6858 


.5 


551.5459 


83.2522 


.6 


401.1500 


71.0000 


.6 


555.7163 


83.5664 


.7 


404.7078 


71.3-142 


.7 


559.9025 


83.8805 


.8 


408.2814 


71.6283 


.8 


564 1044 


84.1947 


.9 


411.6707 


71.9425 


.9 


568.3220 


84.5088 


23.0 


415.4756 


72.2566 


27.0 


572.5553 


84.8230 


.1 


419.0993 


72.5708 


.1 


576.8043 


85.1372 


.2 


422.7327 


72.8849 


.2 


581.0890 


85.4513 


.3 


426.3848 


73.1991 


.3 


585.3494 


85.7655 


.4 


430.0526 


73.5133 


.4 


589.6455 


86.0796 


.5 


433.7361 


73.8274 


.5 


593.9574 


86.3938 


.6 


487.4354 


74.1416 


.6 


598.2849 


86.7080 


.7 


441.1508 


74.4557 


.7 


602.6282 


87.0221 


.8 


444.8809 


74.7699 


.8 


606.9871 


87.3363 


.9 


448.6273 


75.0841 


.9 


611.3618 


87.6504 


24.0 


452.3893 


75.3982 


28.0 


615.7522 


87.9646 


.1 


456.1671 


75.7124 


.1 


620.1582 


88.2788 


.2 


459.9606 


76.0265 


.2 


624.5800 


88.5929 


.3 


463.7698 


76.3407 


.3 


629.0175 


88.9071 


.4 


467.5947 . 


76.6549 


.4 


633.4707 


89.2212 


.5 


471.4352 


76.9690 


.5 


637.9397 


89.5354 


.6 


475.2916 


' 77.2832 


.6 


642.4243 


89.8495 


.7 


479.1636 


77.5973 


.7 


646.9246 


90.1637 


.8 


483.0513 


77.9115 


.8 


651.4407 


90.4779 


.9 


486.9547 


78.2257 


.9 


655.9724 


90.7920 


95.0 


490.8739 


78.5398 


29.0 


660.5199 


91.1063 


.1 


494 8087 


78.8540 


.1 


665.0830 


91.4203 


.2 


498.7592 


79.1681 


.2 


669.6619 


91.7345 


.3 


603.7255 


79.4823 


.3 


674 2565 


92.0487 


.4 


506.7075 


79.7905 


.4 


67*.8668 


92.3628 


.5 


5 1 0.7052 


80,1106 


.5 


683.4928 


92.6770 


.6 


514.7185 


80.4248 


.6 


688.1345 


92.9911 


.7 


518.7478 


HO. 7389 


.7 


692.7919 


93.3053 


.8 


522 7924 


81.0581 


.8 


697.4650 


93.6195 


.9 


52(5.8529 


81.8672 


.9 


702.1538 


93.9336 



n8 



Hand Book of Calmlations. 



TA BLE— (Continued.) 

CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OIT CHICLES. 



706.8583 
711.5786 
716.3145 
721.0662 
725.8336 

730.6167 
735.4154 
740.2299 
745.0601 
749.9060 

754.7676 
759.6450 

764.5380 
769.4467 

774.3712 

779.3113 

784.2672 
789.2388 
794.2260 
799.2290 

804.2477 
809.2821 
814 3322 
819.39f0 
824.47S6 

839.5768 
834.6898 
839.8185 
814.9628 
850.1229 

855.2986 
860.4902 
865.6973 

870.9202 
876.15S8 

881.4131 
886.6831 
891.9688 
897.2703 
902.5874 



Circum. 



94.2478 
94.5619 
94.8761 
95.1903 
95.5044 

95.8186 
96.1327 
96.4469 
96.7611 
97.0752 

97.3894 

97.7035 
98.0177 
98.3319 
98.6460 

98.9602 
99.2743 

99.5885 

99.9026 

100.2168 

100.5310 
100.8451 
101.1593 
101.4734 
101.7876 

102.1018 
102.4159 
102.7301 
103 0442 
103 3584 

103.6726 
103.9867 
104.3009 
104.6150 
104.9292 

105.2434 
105.5575 
105.8717 
106.1858 
106.5000 



Diam. 



34.0 
.1 
.2 
.3 

.4 

.5 
.6 

.7 
.8 
.9 

35.0 

.1 
.2 



.5 
.6 

.7 
.8 
.9 

36.0 

!£ 

.3 

.4 

.5 
.6 

.7 
.8 
.9 

S7.0 
.1 
.2 
.3 

• 4 

.5 
.6 

.7 
.8 
.9 



Area. 


Circum. 


907.9203 


106.8142 


913.2688 


107.1283 


918.6331 


107.4425 


924.0131 


107.7566 


929.4088 


108.0708 


334.8202 


108.3849 


940.2473 


108.6991 


945.6901 


109.0133 


951.1486 


109.3274 


956.6228 


109.6416 



962.1128 
967.6184 
973.1397 

978.6768 
984.2296 

989.7980 

995.3822 

1000.9821 

1006.5977 

1012.2290 

1017.8760 
1023.5387 
1029.2172 
1034.9113 
1040.6212 

1046.3467 
1052. 0880 
1057.8449 
1063 6 1 76 
10*19.4060 

1075.2101 
1081.0299 
1386.8654 
1092.7166 

1098X835 

1104.4662 
1110.3645 
1116.2786 
1122.2083 
1128.1538 



109.9557 
110.2699 
110.5841 
110.8982 
111.2124 

111.5265 
111.8407 
112.1549 
112.4690 
112.7832 

113.0973 
113.4115 
113.7257 
114.0398 
114.3540 

114.0681 
114.9823 
115.2965 
115.6106 
115.9248 

11C. 2389 
116.5531 
116.8672 
117.1814 
117.4956 

117.8097 
118.1239 
118.4380 
118.7522 
119.0664 



Hand 



of Calculations. 



"3 



TA BLE— {Continued. ) 

CONTAINING THE DIAMETERS. CIRCUMFERENCES AND AREAS OF CIRCLES. 



Diain. 


Area. 


Ctrcum. 


IMam. 


Area. 


Circum. 


38.0 

.1 
o 

A 


1134.1149 
1140.0918 
1146.0844 
1152.0927 

1158.1167 


119.3805 
119.6947 
120.0088 
120.3230 
! 120.6372 


42.0 
.1 
.2 
.3 
A 


1385.4424 
1392.0476 
1398.6685 
1405.3051 
1411.9574 


131.9469 
132.2611 
132.5752 
132 8894 
133.2035 


.5 
.6 

'.8 
.9 


1164.1564 
1170.2118 

1170/2830 
1182.3698 
1188.4724 


120.9513 
121.2655 
121.5796 
121.8938 
122.2080 


.5 
.6 

.7 
.8 
.9 


1418.6254 
1425.3092 
1432.0086 
1438.7238 
1445.4546 


133.5177 
133.8318 
134.1460 
134.4602 
134.7743 


39.0 
.1 
.2 
.3 

A 


1194.5906 
1200.7246 
1206.8742 
1213.0396 
1219.2207 


122.5221 
122.8363 
123.1504 
123.4646 

123.7788 


43.0 
.1 

.2 
.3 
.4 


1452.2012 
1458.9635 
1465.7415 
1472.5352 
1479.3446 


135.0885 
135.4026 
135.7168 
136.0310 
136.3451 


.5 
.6 

.: 

.8 ' 
.9 


1225.4175 
1231.6300 
1237.8582 
1244.1021 
1250.3617 


124.0929 
124.4071 
124.7212 
125.0354 
125.3495 


.5 
.6 

.7 
.8 
.9 


1486.1697 
1493.0105 
1499.8670 
1506.7393 
1513.6272 


136.6593 
136.9734 
137.2876 
137.6018 
137.9159 


40.0 
.1 
.2 
.3 
-4 


1256.6371 
1262.9281 
1269.2348 
1375.5573 

1281.8955 


125.6637 
125.9779 
126.2920 
126.6062 
128.9203 


44.0 
.1 
.2 
.3 
.4 


1520.5308 
1527.4502 
1534.3853 
1541.3360 
1548.3025 


138.2301 
138.5442 
138.8584 
139.1726 
139.4867 


.5 
.6 
.7 

.8 
.9 


1288.2493 
1294.3189 
1301.0042 
1307.4052 
1313.8219 


127.2345 

127.5487 
127.8628 
128.1770 
128.4911 


.5 
.6 
.7 
.8 
.9 


1555.2847 
1562.2826 
1569.2962 
1576.3255 
1583.3706 


139.8009 
140.1153 
140.4292 
140.7434 
141.0575 


41.0 
.1 
.2 
.3 
.4 


1320.2548 
1326.7084 

1333.1603 
1339.6458 

1346.1410 


128.8058 
129.1195 
129.4336 
129.7478 
130.0619 


43.0 

.1 
.2 

.3 
.1 


1590.4313 
1597.5077 
1604.5999 
1611.7077 
1618.8313 


141.3717 
141.6858 
142.0000 
142.3142 
142.6283 


.5 

.6 

.7 
.8 
.9 


1352.0520 
1359.1786 
1365.7210 

1372.2701 
1378.8529 


130.3761 
130.0008 
131.0044 
131. 31 SO 
131.6221 


,5 

.0 
.7 
.8 
.9 


1025.0705 
1683.1255 

1040/ 002 
1647.4826 
1054.0847 


142.9425 
143.2566 
143.5708 
143.8849 
111.1991 



120 



Hand Book of Calculations, 



TABLE— (Continued.) 

CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES. 



Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 


46.0 
.1 

.2 
.3 
.4 


1661.9025 
1669.1360 
1676.3853 
1683.6502 
1690.9308 


144.5133 
144.8274 
145.1416 
145.4557 
145.7699 


50.0 
.1 

.2 

.3 

.4 


1963.4954 
1971.3572 
1979.2348 
1987.1280 
1995.0370 


157.0796 
157.3938 
157.7080 
158.0221 
158.3363 


.5 
.6 

.7 
.8 
.9 


1698.2272 
1705.5392 

1712.8670 
1720.2105 
1727.5697 


146.0841 
146.3982 
146.7124 
147.0265 
147.3407 


.5 

.6 

.7 
.8 
.9 


< 2002.9617 
2010.9020 
2018.8581 
2026.8299 
2034.8174 


158.6504 
158.9646 
159.2787 
159.5929 
159.9071 


47.0 
.1 
.2 
.3 
.4 


1734.9445 
1742.3351 
1749.7414 
1757.1635 
1764.6012 


147.6550 
147.9690 

148.2832 
148.5973 
148.9115 


51.0 
.1 
.2 
]3 
.4 


2042.8206 
2050.8395 
2058.8742 
2066.9245 
2074.9905 


160.2212 
160.5354 
160.8495 
161.1637 
161.4779 


.5 
.6 

.7 
.8 
.9 


1772.0546 
1779.5237 
1787.0086 
1794.5091 
1802.0254 


149.2257 
149.5398 
149.8540 
150.1681 
150.4823 


.5 
.6 

.7' 
.8 
.9 


2083.0723 
2091.1697 
2^99.2829 
2107.4118 
2115.5563 


161.7920- 
162.1062 
162.4203 
162.7345 
163.0487 


48.0 
.1 
.2 
.3 
.4 


1809.5574 
1817.1050 
1824.6684 
1832.2475 
1839.8423 


150.7984 
151.1106 
151.4248 
151.7389 
152.0531 


52.0 
.1 
.2 
.*3 
.4 


2123.7166 
2131.8926 
2140.0843 
2148.2917 
2156.5149 


163.3628 
163.6770 
163.9911 
164.3053 
164.6195 


.5 
.6 

.7 
.8 
.9 


1847.4528 
1855.0790 
1862.7210 
1870.3786 
1878.0519 


152.3672 
152.6814 
152.9956 
153.3097 
153.6239 


,5 
.6 
.7 
.8 
.9 


2164.7537 
2173.0082 

2181.2785 
2189.5644 
2197.8661 


164.9336 
165.2479 
165.5619 
165.8761 
166.190& 


49.0 
.1 
.2 
.3 
.4 


1885.7409 
1893.4457 
1901.1662 
1908.9024 
1916.6543 


153.9380 
154.2522 
154.5664 
154.8805 
155.1947 


53.0 
.1 
.2 
.3 
.4 


2206.1834 
2214.5165 
2222.8653 
2231.2298 
2239.6100 


166.5044 
166.8186 
167.1327 
167.4469 
167.7610 


.5 

.6 

.7 
.8 
.9 


1924.4218 
1932.2051 
1940.0042 
1947. 818 J 
1955.6493 


155.5088 
155.8230 
156.1372 
156.4513 
156.7655 


.5' 

.6 
.7 
.8 
.9 


2^48.0059 
2256.4175 

2264.8448 
2273.2879 
2281 7466 ! 


168.0752 
168.3894 
168.7035 
169.0177 
169.3318 



Hand Book of Calculations. 



121 



TABLE— (Continued.) 

CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES. 



Diain. 


Area. 


Circum 


Diain. 


Area. 


54.0 
.1 
t 2 

;3 

A 


2290.2210 
2298.7112 
2307.2171 
2315.7386 
23:4.2759 


169.6460 
169.9602 
170.2743 
170.5885 
170.9026 


58.0 
.1 
.2 
.3 
.4 


2642.0794 
2651.1979 
2660.3321 
2669.4820 

2678.6476 


.5 
.6 
.7 
.8 
.9 


2332.8289 
2341.3976 
2:549.9820 
2258.5821 

2867.1979 


171.2168 
171.5ol0 
171.8451 
172.1593 
172.4735 


.5 
.6 

.7 
.8 
.9 


2687.8289 
2697.0259 
2706.2386 
2715.4670 
2724.7112 


55.0 
.1 
.2 
.3 
.4 


2375.8294 
2384.4767 
2393.1396 
2401.8183 
2410.ol26 


172.7876 
173.1017 
173.4159 
173.7301 
174.0442 


59.0 
.1 
.2 
.3 
.4 


2733.9710 
2743.2466 

2752.5378 
2761.8448 
2771.1675 


.5 
.6 
.7 

.8 
.9 


2419.2227 
2427.9485 
2436.6899 
2445.4471 
2454.2200 


174.3584 
174.6726 
174.9867 
175.3009 
175.6150 


.5 
.6 

.7 
.8 
.9 


2780.5058 
2789.8599 
2799.2297 
2808.6152 
2818.0165 


56.0 
.1 
.2 
.3 
A 


2463.0080 
2471.8130 
2480.6330 
2489.4687 
2498.3201 


175.9292 
176.2433 

176.5575 
176.8717 
177.1858 


60.0 
.1 
.2 
.3 
.4 


2827.4334 
2836 8660 
2846.3144 

2855.7784 
2865.2582 


.5 
.6 

.7 
,8 

.9 


2507.1873 
2516.0701 
2524.9687 
2533.8830 

2542.8129 


177.5000 
177.8141 
178.1283 
178.4125 
178.7566 


. > 

.7 
.8 
.9 


2874.7536 
2884.2648 
2803.7917 
2903.3343 
2912.8926 


57.0 
.1 
.2 
.3 
.4 


2551.7586 

256.). 7200 
2569.6071 
2578.6809 

2587.6085 


179.070.8 
170.3849 
179 6991 
180.01.;.; 
180.3274 


61.0 
.1 
.2 
.3 
.4 


2922.4666 
293^.0563 
2941.6617 
2951.2828 
2960.9197 


.5 

.<; 

.7 

.8 
.9 


2590.7227 
2605.7626 
2614.8183 
2623.8896 
2632.0767 


180.6416 
180.9557 
1M.2699 
181.5841 
181.8982 


.5 
.6 

.7 
.8 
.9 


2970.5722 
2980.2405 
2989.9244 
2999.6241 
3009.3395 



182.2124 
182.5265 

182.8407 
183.1549 
183.4690 

183.7832 
184.0973 
184.4115 
184.7256 
185.0398 

185.3540 
185.6681 
185.9823 
186.2964 
186.6106 

186.9248 
187.2389 
187.5531 
187.8672 
188.1814 

188.4956 
188.8097 
189.1239 
189.4380 
189.7522 

190.0664 
190.3805 
190.6947 
191.0088 
191.3230 

191.6372 
191.9513 
192.2655 
192.5796 
192.8938 

193.2079 
193.5221 
193.8363 
191.1504 
194.4646 



122 



Hand Book of Calculations, 



TABLE— (Continued.) 

CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OP CIRCLES. 



Diam. 


Area. 


Circum. 


Diam 


Area. 


Circum. 


62 

.1 
.2 
.3 
.4 


3019.0705 
3028.8173 
3038.5798 
3048.3580 
3058.1520 


194.7787 
195.0929 
195.4071 
195.7212 
196.0354 


66.0 
.1 

.2 
.3 

.4 


3421.1944 
3431.5695 
3441.9603 
3452.3669 
3462.7891 


207.3451 
207.6593 
207.9734 

208.2876 
208.6017 


.5 

.6 

.7 
.8 
.9 


3067.9616 

3077.7869 
3087.6279 
3097.4847 
3107.3571 


196.3495 
196.6637 
196.9779 
197.2920 
197.6062 


.5 
.6 

.7 
.8 
.9 


3473.2270 
3483.6807 
3494.1500 
3504.6351 
3515.1359 


208.9159 
209.2301 
209.5442 
209.8584 
210.1725 


63.0 
.1 
.2 
.3 

.4 


3117.2453 
3127.1492 
3137.0688 
3147.0040 
3156.9550 


197.9203 

198.2345 
198.5487 
198.8628 
199.1770 


67.0 
.1 
.2 
.3 

.4 


3525.6524 
3536.1845 
3546.7324 
3557.2960 
3567.8754 


210.4867 
210.8009 
211.1150 
211.4292 
211.7433 


.5 
.6 

.7 
.8 
.9 


3166.9217 
3176.9043 
3186.9023 
3196.9161 
3206.9456 


199.4911 
199.8053 
200.1195 
200.4336 

200.7478 


.5 
.6 

.7 
.8 
.9 


3578.4704 
3589.0811 
3599.7075 
3610.3497 
3621.0075 


212.0575 
212.3717 
212 6858 
213.0000 
213.3141 


£4.0 
.1 

.2 
.3 
.4 


3216.9909 

3227.0518 
3237.1285 
3247.2222 
3257.3.89 


201.0620 
201.3761 

201.6902 
202.0044 

202.3186 


68.0 
.1 
.2 
.3 
.4 


3631.6811 
3642.3704 
3653.0754 
3663.7960 
3674.5324 


213.6283 
213.9425 
214.2566 
214.5708 
214.8849 


.5 
.6 

.7 
.8 
.9 


3267.4527 
3277.5922 

3287.7474 
3^97.9183 
3308. 1049 


202.6327 
202.9469 
203.2610 
203.5752 

203.8894 


.5 
.6 

.7 
.8 
.9 


3685.2845 
3696.0523 
3708.8359 
3717.6351 
3728.45U0 


215.1991 
215.5133 

215.8274 
216.1416 
216.4556 


€5.0 
.1 
.2 
.3 

.4 


3318.3072 
3328.5253 
3333.7590 
3349.0085 
3359.2736 


204.2035 
204.5176 
204.8318 
205.1460 
205.4602 


69.0 
.1 
.2 
.3 

.4 


3739.2807 
3750.1270 
3760.9891 
3771.8668 
3782.7603 


216.7699 
217.0841 
217.3982 
217.7124 
218.0265 


.5 
.6 

.7 
.8 
.9 


3369.5515 
3379.8510 
3390.1633 
3400.4913 
3410.8350 


205.7743 

206.0885 
206.4026 
206.7168 
207.0310 " 8 


.5 

i 

.9 


3793.6695 
3804.5944 
3815.5350 
3826.4913 
3837.4633 


218.3407 
218.6548 
218.9690 
219.2832 
219.5973 



Hand Book of Calculations. 



123 



TABLE— (Continued.) 



•CONTAINING THE DIAMETERS. CIRCUMFERENCES AND AREAS OF CIRCLES, 



Diani. 


Area. 


CIrcum. 


Diani. 


Area. 


Circum. 


70.0 
.1 
.2 
.3 
.4 


3848.4510 
3.-59.4544 
3870.4736 
3881.5084 
3892.5590 


219.9115 
220.2256 
220.5398 
! 220.8540 
221.1681 


74.0 
.1 
.2 
.*3 
.4 


4300.8403 
4312.4721 
43^4.1195 
4335.7827 
4347.4616 


232.4779 
232.7920 
233.1062 
233.4203 
233.7345 


.5 

.6 

.7 
.8 
.9 


3903.6252 
3914.7072 
3925.8049 

3936.9182 

3948.0473 


221.4823 
221.7964 
222.1106 
222.4248 
222.7389 


.5 
.6 

.7 
.8 
.9 


4359.1562 
4370.8664 
4382.5924 
4394.3341 

4406.0916 


234.0487 
234.3628 
234.6770 
234.9911 
235.3053 


71.0 
.1 

.4 


3959.1921 
3970.3526 

3981.5289 
3992.7208 
4003.9284 


223.0531 
223.3672 
223.6814 
223.9956 
224.3097 


75.0 
.1 
.2 
.3 
.4 


4417.8647 
4429.6535 
4441.4580 
4453.2783 
4465.1142 


235.6194 
235.9336 
236.2478 
236.5619 
236.8761 


.5 

.6 
.7 
.8 
.9 


4015.1518 
4026.3908 
4037.6456 
4048.9160 
4060.2022 


224.6239 
224.9380 
225.2522 
225.5664 

225.8805 


.5 

.6 

.7 
.8 
.9 


4476 9659 
4488.8332 
4500.7163 
4512.6151 
4524.5296 


237.1902 
237.5044 

237.8186 
238.1327 
238.4469 


72.0 
.1 
.•J 
.3 
.4 


4071.5041 
4082.8217 
4094.1550 
4105.5040 
4116.8687 


226.1947 

226.5088 
226.8230 
227.1371 
227.4513 


76.0 

i 
i 


4536.4598 

4548.4057 
4560.3673 
4572.3446 

4584. 3 i77 


238.7610 
239.0752 
239.3894 
239.7035 
240.0177 


.5 

.6 
.7 
.8 
.9 


4128.2491 

4139.0452 
4151.0571 
4162.4846 
4173.9279 


227.7655 
228.0796 J 

228.3938 
228.7079 

229.0221 


.5 

.6 

.7 
.8 
.9 


4596.3464 
4608.3708 
4620.4110 
4632.4669 
4344.5384 


240.3318 
240.6460 
240.9602 
241.2743 

241.5885 


73.0 
.1 
.2 
.3 

.4 


4185.3868 
4196.8615 
4208.3519 
4219.8579 
4281.3797 


229.3363 
229.6504 

229.9646 
230.2787 
230.5929 


77.0 
.1 
.2 
.3 
.4 


4656.6257 

4668.7287 
4680.8474 
4692.9818 
4705.1319 


241.9026 
242.2168 
242.5310 
242.H451 
243.1592 


.5 

.6 

.7 
.8 
.9 


4242.9172 
4254 4701 
1266.0394 
4277.6240 

4289.2243 


230.9071 
231.2212 

231.5354 
281.H395 

232.1037 


.5 
.0 

.7 
.8 
.9 


4717.2977 
4729.4792 
4741.6765 
475:', 8894 
4766.1 LSI 


243.4734 
243.7876 
244.1017 
244.4159 
244.7301 



124 



Hand Book of Calculations. 



TABLE— (Continued.) 

CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES, 



Dlani. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 


78.0 
.1 
.2 
.3 
.4 


4778.3624 
4790.6225 
4802.8983 
4815.1897 
4827.4969 


245.0442 
245.3584 
245.6725 
245.9867 
246.3009 


82.0 
.1 
.2 
.'3 
.4 


5281.0173 
5293.9056 
5306.8097 
5319.7295 
5332,6650 


257.6106 
257.9247 

258.2389 
258.5531 

258.8672 


.5 
.6 

.7 
.8 
.9 


4839.8189 
4852.1584 
4864.5128 
4876.8828 
4889.2685 


246.6150 
246.9292 
247.2433 

247.5575 
247.8717 


.5 
.6 

.7 
.8 
.9 


5345.6162 

5358.5832 
5371.5658 
5384.5641 

5397.5782 


259.1814 
259.4956 
259.8097 
260. 1239 
260.4380 


79.0 
.1 
.2 
.3 
.4 


4901.6699 
4914.0871 
4926.5199 
4938.9685 
4951.4328 


248.1858 
248.5000 
248.8141 
249.1283 
249.4425 


83.0 
.1 
.2 
.3 
.4 


5410.6079 
5423.6534 
5436.7146 
5449.7915 

5462.8840 


260.7522 
261.0663 
261.3805 
261.6947 

262.0088 


.5 
.6 

.7 
.8 
.9 


4963.9127 
4976.4084 
4988.9198 
5001.4469 
5013.9897 


249.7566 

250.0708 
250.3850 
250.6991 
251.0133 


.5 
.6 

.7 
.8 
.9 


5475.9923 
5489.1163 
5502.2561 
5515.4115 

5528.5826 


262.3230 
262.6371 
262.9513 
263.2655 
263.5796 


80.0 
.1 
.2 
.3 
.4 


5026.5482 
5039.1225 
5051.7124 
5064.3180 
5076.9394 


251.3274 
251.6416 
251.9557 

252.2699 
252.5840 


84.0 
.1 
.2 
.3 

.4 


5541.7694 
5554.9720 
5568.1902 
5581.4242 
5594.6739 


263.893$ 
264.2079 
264.5221 
264.8363 
265.1514 


.5 
.6 

.7 
.8 
.9 


5089.5764 
5102.2292 
5114 8977 
5127.5819 
5140.2818 


252.8982 
253.2124 
253.5265 
253.8407 
254.1548 


.5 

.6 

.7 
.8 
.9 


5607.9392 
5621.2203 
5634.5171 
5647.8296 
5661.1578 


265.4646 

265.7787 
266.0929 
266.4071 
266.7212 


81.0 
.1 
.2 
.3 
.4 


5152.9973 

5165.7287 
5178.4757 
5191.2384 
5204.0168 


254.4690 
2*4.7832 
255.0973 
255.4115 
255.7256 


85.0 
.1 
.2 
.3 
.4 


5674.5017 

5687.8614 
5701.2367 

5714.6277 

5728.0345 


267.0354 
267.3495 
267.6637 
267.9779 
268.2920 


.5 
.6 
.7 
.8 
.9 


5216.8110 
5229.6208 
5242.4463 
5255.2876 
5268.1446 


256.0398 
256.3540 
256.6681 
" 256.9823 
257.2966 


.5 
.6 

• .7 
.8 
.9 


5741.4569 

5754.8951 
5768.3490 
5781.8185 
5795.3038 


268.6062 
268.9203 
269.2345 
269.5486 
269.8628 



Hand Book of Calculations. 



™S 



TABLE— {Continued. ) 

CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES. 



Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 


86.0 
.1 

.2 

.3 
.4 


5808.8048 
5822.3215 
5835.8539 
5849.4020 
5862.9659 


270.2770 
270.4911 
270.8053 
271.1194 
271.4336 


90.0 
.1 
.2 
.3 
.4 


6361.7251 
6375.8701 
6390.0309 
6404.2073 
6418.3995 


282.7433 

283.0575 

283.3717 
283.6858 
284.0000 


.5 
.6 

.7 
.8 
.9 


5876.5454 
5890.1407 
5903.7516 
5917.3783 
5931.0206 


271.7478 
272.0619 
272.3761 
272.6902 
273.0044 


.5 
.6 

.7 
.8 
.9 


6432.6073 
6446.8309 
6461.0701 
6475.3251 
6489.5958 


284.3141 

284.6283 
284.9425 
285.2566 
285.5708 


87.0 
.1 
.2 
.3 
.4 


5944.6787 
5958.3525 
5972.0420 

5985.7472 
5999.4681 


273.3186 
273.6327 
273.9469 
274.2610 
274.5752 


91.0 
.1 
.2 
.3 

.4 


6503.8822 
6518.1843 
6532.5021 
6546.8356 
6561.1848 


285.8849 
286.1991 
286.5133 
286.8274 
287.1416 


.5 
.6 

.7 
.8 
.9 


6013.2047 
6026.9570 
6040 72oD 
6054.5088 
6068.3082 


274.8894 
275.2035 
275.5177 
275.8318 
276.1460 


.5 
.6 

.7 
.8 
.9 


6575.5498 
6589.9304 
6604 3268 
6618.7388 
6633.1666 


287.4557 
287=7699 

288.0840 
288.3982 
288.7124 


88.0 
.1 
.2 
.3 
.4 


6082.1234 
8095.9542 
6109.8008 
6123.6631 

6137.5411 


276.4602 
276.7743 
277.0885 
277.4026 
277.7168 


92.0 
.1 
.2 
.3 
.4 


6647.6101 
6662.0692 
6676.5441 
6691.0347 
6705.5410 


289.0265 
289.3407 
289.6548 
289.9690 
290.2832 


.6 

.6 

•• 
.8 
.9 


6151.4348 
6165.3442 
6179.2693 
6193.2101 
6307.1666 


278.0309 
278.3451 
278.6563 
278.9740 

279.2876 


.5 
.6 

.7 
.8 
.9 


6720.0630 
6734.6008 
6749.1542 
6763.7233 

6778.3082 


290.5973 
290.9115 
291.2256 

291.5308 
201.8540 


89.0 
.1 
.2 
.8 

.4 


6221.1389 
6235.1268 
6349.1304 
6263.1498 

6277.1849 


270. GO 17 
279.9159 
280.2301 

280 5442 
,8584 


93.0 
.1 
.2 
.3 
.4 


6792.9087 
6807.5250 
0822.150!) 
6836.8046 
6851.4680 


292.1681 
292.4823 
292.7964 
293.1106 
293.4248 


.6 

i 

.9 


6391.2356 

6305.3021 

6319.3843 

6333. 

6347.5958 


281.1725 
281.4867 
281.8009 
282.1150 

282.4292 


.5 
,6 

i 

.9 


6866.1471 
6880.8419 
6895.5524 
6910.2786 

0025.0205 


293.7389 
294.0531 
294.3072 
294. OH 14 
294.9956 



126 



Hand Book of Calculations. 



TABLE— (Concluded. ) 

CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES* 



Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 


94.0 


6939.7782 


295.3097 


97.0 


7389.8113 


304.7345 


.1 


6954.5515 


295.6239 


.1 


7405.0559 


305.0486 


.2 


6969.3106 


295.9380 


.2 


7420.3162 


305.3628 


.3 


6984.1453 


296.2522 


.3 


7435.5922 


305.6770 


.4 


6998.9658 


296.5663 


.4 


7450.8839 


305 9911 


.5 


7013.8019 


296.8805 


.5 


7466.1913 


306.3053 


.6 


7028.6538 


297.1947 


.6 


7481.5144 


306.6194 


.7 


7043.5214 


297.5088 


.7 


7496.8532 


306.9336 


.8 


7058.4047 


297.8230 


.8 


7521.2078 


307.2478 


.9 


7073.3033 


298.1371 


.9 


7527.5780 


307.5619 


95.0 


7088.2184 


298.4513 


98.0 


7542.9640 


307.8761 


.1 


7103.1488 


298.7655 


.1 


7558.3656 


308.1902 


.2 


7118.1950 


299.0796 


.2 


7573.7830 


308.5044 


.3 


7133.0568 


299.3938 


.3 


7589.2161 


308.8186 


.4 


7148.0343 


299.7079 


.4 


7604.6648 


309.1327 


.5 


7163.0276 


300.0221 


.5 


7620.1293 


309.4469 


.6 


7178.0366 


300.3363 


.6 


7635.6095 


309.7610 


.7 


7193.0612 


300.6504 


.7 


7651.1054 


310.0752 


.8 


7208.1016 


300.9646 


.8 


7666.6170 


310.3894 


.9 


7223.1577 


301.2787 


.9 


7682.1444 


310.7035 


96.0 


7238.2295 


301.5929 


99.0 


7697.6893 


311.0177 


.1 


7253.3170 


301.9071 


.1 


7713.2461 


311.3318 


.2 


7268.4202 


302.2212 


.2 


7728.8206 


311.6460 


.3 


7283.5391 


302.5354 


.3 


7744.4107 


311.9602 


.4 


7298.6737 


302.8405 


.4 


7760.0166 


312.2743 


.5 


7313.8240 


303.1637 


.5 


7775.6382 


312.5885 


.6 


7328.9901 


303.4779 


.6 


7791.2754 


312.9026 


.7 


7344.1718 


303.7920 


.7 


7806.9284 


313.2168 


.8 


7359.3693 


304.1062 


.8 


7822.5971 


313.5309 


.9 


7374.5824 


304.4203 


.9 


7838.2815 


313.8451 








100.0 


7853.9816 


314.1593 




Hand Book of Calculations. I2J 



GEOMETRY. 



Geometry is one of the oldest and simplest of sciences; it may 
be defined as the science of measurement; Mensuration as 
already briefly outlined in this work, belongs properly under 
this division. 

Geometry is the root from which all regular mathematical 
calculations issue. It has claimed the best thought of practi- 
cal men from the times of the Greeks and Eomans two thous- 
and years ago; they derived their knowledge of the science 
from the Egyptians, who in turn were indebted to the Chaldeans 
and Hindoos in times beyond any authentic history; hence it 
was under the operations of the laws explained in geometry, that 
the pyramids of Egypt and the temples of Greece, were con- 
structed, as well as the engines of war and appliances of peace 
of ancient times. 

The elementary conceptions of geometry are few. 
1. A point. 

2. A line. 

3. A surface. 

4. A solid, and 

5. An angle. 

From these definitions, as data, a vast number of mathemati- 
cal calculations have been deduced ; of which a few of the 
most; elementary will be explained and illustrated in this work; 
but these few will repay the attention of the student as the 
mutual relation between practical engineering and geometry is 
very intimate indeed — as will be apparent. 



128 Hand Book of Calculations. 

GEOMETKICAL DEFINITIONS. 

A point is mere position, and has no magnitude. 

A line is that which has extension in length only. The 
extremities of lines are points. 

A surface is that which has extension in length and breadth 
only. 

A solid is that which has extension in length, breadth and 
thickness. 

An angle is the difference in the direction of ^^ 
two lines proceeding from the same point. ^^ 

Lines, Surfaces, Angles and Solids constitute the different 
kinds of quantity called geometrical magnitudes. 

Parallel lines are lines which have the same — 

direction; hence parallel lines can never meet, — 

however far they may be produced; for two lines taking the 
same direction cannot approach or recede from each other. 

An Axiom is a self-evident truth, not only too simple to 
require, but too simple to admit of demonstration, 

A Proposition is something which is either proposed to be 
done, or to be demonstrated, and is either a problem or a 
theorem. 

A Problem is something proposed to be done. 

A Theorem is somethi tig proposed to be demonstrated. 

A Hypothesis is a supposition made with a view to draw 
from it some consequence which establishes the truth or false- 
hood of a proposition, or solves a problem. 

A Lemma is something which is premised, or demonstrated, 
in order to render what follows more easy. 

A Corollary is a consequent truth derived immediately from 
some preceding truth or demonstration. 

A Scholium is a remark or observation made upon something 
going before it. 

A Postulate is a problem, the solution of which is self-evident. 



Hand Book of Calculations. 



I2Q 



Examples of Postulates. 
Let it be granted — 

I. That a straight line can be drawn from any one point to 
any other point: 

II. That a straight line can be produced to any distance, or 
terminated at any point; 

III. That the circumference of a circle can be described 
about any center, at any distance from that center. 

ABBREVIATIONS. 

The common algebraic signs are used in Geometry, and it is 

necessary that the student in geometry should understand some 

of the more simple operations of algebra. As the terms circle, 

angle, triangle, hypothesis, axiom, theorem, corollary, and 

definition are constantly occurring in a course of geometry, 

they are abbreviated as shown in the following list: 

Addition is expressed by . . . . -j- 

Subtraction - 

Multiplication " " . . . .X 

Equality and Equivalency are expressed by = 

Greater thari. is expressed by . . . , > 

Less than. st " . . < 

Thus B is greater than A, is written . . B> A 

B is less than A " . . . B<A 

A circle is expressed by . . . . O 

An angle " " . . . . L 

A right angle is expressed by . . . R. L 

Degrees, minutes and seconds are expressed by 

A triangle is expressed by . . . . .a 

The term Hypothesis is expressed by . . . (Ily.) 

Axiom " . (Ax.) 

Theorem < (Th.) 

irollarj " " . . . (Cor.) 

Definition •■"... (Dei) 

Perpendicular is expressed by . . j_ 

The difference of two quantities, when it is not known 

which is the greater, is expressed by the symbol . ~ 

Thus, the difference between A and B is written A ^ B. 



ijo Hand Book of Calculations. 



AXIOMS. 

1. Tilings which are equal to the same thing are equal to each 
other. 

2. When equals are added to equals the tuhole are equal. 

3. When equals are taken from equals the remainders are 
equal. 

4. When equals are added to unequals the wholes are unequal. 

5. When equals are taken from, unequals the remainders are 
unequal. 

6. Things which are double of the same thing, or equal things 
are equal to each other. 

7. Things which are halves of the same thing, or of equal 
things, are equal to each other. 

8. The whole is greater than any of its parts. 

9. Every whole is equal to all its parts taken together. 

10. Things which coincide, or fill the same space, are identi- 
cal, or mutually equal in all their parts. 

11. All right angles are equal to one another. 

12. A straight line is the shortest distance between two points. 

13. Two straight lines cannot inclose a space. 

ANGLES. 

To make an angle apparent, the two lines 
must meet in a point, as AB and AC, which 
meet in the point A. A« 



Fig. 43. 
Angles are measured by degrees. 

A Degree is one of the three hundred and sixty equal parts of 
the space about a point in a plane. 

Angles are distinguished in respect to magnitude by the term 
Right, Acute and Obtuse Angles. 

A Right A ngle is that formed by one line 

meeting another, so as to make equal angles 

with that other. — — — — — 

Fig. 44. 

The lines forming a right angle are perpendicular to each 

other. 



Hand Book of Calculations. iji 



An Acute Angle is less than a right angle. 



Fig. 45. 



An Obtuse Angle is greater than a 
right angle. 

Obtuse and acute angles are also called Fig. 46. 

oblique angles; and lines which are neither parallel nor perpen- 
dicular to each other are called oblique lines. 

The Vertex or Apex of an angle is the point in which the 
including lines meet. 

An angle is commonly designated by a letter at its vertex; 
but when two or more angles have their vertices at the same 
point, they cannot be thus distinguished. 



D 




For example, when the three lines 
AB, A C, and AD meet in the common 
point A, we designate either of the 
angles formed, by three letters, plac- A 
ing that at the vertex between those 
at the opposite extremities of the in- 
cluding lines. Thus, we say, the 
angle BAC, etc. 



PLANE FIGURES. 

A Plane Figure, in geometry, is a portion of a plane bounded 
straight or curved lines, or by both combined. 

A Polygon is a plane figure bounded by straight lines called 
tin -ides of the polygon. The least number of sides that can 
bound a polygon is three. 

FIGURES OF THREE SIDES. 
A Triangle is a polygon having three sides and three angles. 
TH is a Latin prefix signifying three; hence a Triangle is liter- 
ally a figure containing three angles. 

A Scalene Triangle \± on< in which no two 
Hides are equal. 

Fig. 48. 




IJ2 Hand Book of Calculations. 



An Isosceles Triangle is one in which two 
of the sides are equal. 



An Equilateral Triangle is one in which 
the three sides are equal. 



A Right-Angled Triangle is one which has 
one of the angles a right angle. 



An Obtuse-Angled Triangle is one having 
an obtuse angle. 




Fig. 49. 




Fig. 50. 



Fig. 51. 




Fig. 52. 



An Equiangular Triangle is one having 
its three angles equal. 




Fig. 53. 




An Acute- Angled Triangle is one in which 
each angle is acute. 

Fig. 54. 
Equiangular triangles are also equal sided, and vice versa. 



Hand Book of Calculations. 



'33 



FIGURES OF FOUR SIDES. 



A Quadrilateral is a polygon having four sides and four 
angles. 



A Parallelogram is a quadrilateral which / / 

bas its opposite sides parallel. Z / 

Fig. 55. 



A Rectangle is a parallelogram naving its 
angles right angles. 

A Square is an equilateral rectangle. 



Fig. 56. 



A Rhomboid is an oblique-angled parallelogram. 



A Rhombus is an equilateral rhomboid. 



A Trapezium is a quadrilateral having no 
two sides parallel. 




Fig 5^ 



Kig. 58. 



A Trapezoid is a quadrilateral in which 
two opposite sides are parallel, and the other 
two oblique. 



Fig. 59. 



Polygons bounded by a greater number of sides than four are 
denominated only by the number of sides. A polygon of five 
sides is called a Pentagon; of six. a Hexagon; of seven, a Hep- 
tagon; of eight, an Octagon, of nine, a Nonagon, etc. 



IJ4 Hand Book of Calculations, 




Diagonals of a polygon are lines joining the 
yertices of angles not adjacent. 

Fig. 60, 
The Perimeter of a polygon is its boundary considered as a 
whole. 

The Hase of a polygon is the side upon which the polygon is 
supposed to stand. 

The Altitude of a polygon is the perpendicular distance 
between the base and a side or angle opposite the base. 



THE CIRCLE. 

A Circle is a plane figure bounded by one 
uniformly curved line ? all of the points in q 
which are at the same distance from a certain 
point within, called the Center. 

The Circumference of a circle is the curved 
line that bounds it. Fig. 61. 

The Diameter of a circle is a line passing through its center, 
and terminating at both ends in the circumference. 

The Radius of a circle is a line extending from its center to 
any point in the circumference. It is one half of the diameter. 
All the diameters of a circle are equal, as are also all the radii. 

An Arc of a circle is any portion of the circumference. 

An angle having its vertex at the center of a circle is meas- 
ured by the arc intercepted by its sides. Thus, the arc AB 
measures the angle A OB; and m general, to compare different 
angles, we have but to compare the arcs, included by their 
sides, of the equal circles having their centers at the vertices of 
ttie angles. 




Hand Book of Calculations, 



*35 



THE FIVE GEOMETRICAL SOLIDS. 

There are five regular solids which are shown in Figs. 62, 63, 
64, 05, and 60. A regular solid is bounded by similar and 
regular plane figures. 




Fig. 62. Fig. 63. Fig. 64. Fig. 65. Fig. 66. 

The tetrahedron, bounded by four equilateral triangles. 

The hexahedron, or cube, bounded by six squares. 

The octahedron, bounded by eight equilateral triangles 

The dodecahedron, bounded by twelve pentagons. 

The icosahedron, bounded by twenty equilateral triangles. 

To find the surface and the cubic contents of any of the five 
regular solids. 

Rule. 

For the surface, multiply the tabular area below, by the 
square of the edge of the solid. 

For the contents, multiply the tabular contents below, by 
the cube of the given edge. 

Regular solids may be circumscribed by spheres, and spheres 
may be inscribed in regular solids. 

m i;ia- E8 anii Cubic Contents of Regular Solids. 



Number 

\ ides. 


NAME. 


A-i'ji. 
Edge-1. 


Contents. 
Edge=l. 


4 


Tetrahedron 


1.7320 


0.1178 




Hexahedron 


6.0000 


1.0000 


8 


Octahedron 


3.4041 


0.47L4 


12 


Dodecahedron .... 


20.0458 


7.6631 


20 


Icosahedron 


8.GG03 


2.1817 



jj6 Hand Book of Calculations. 



PLANE TRIGONOMETRY. 

Trigonometry is that portion of geometry which has for its 
object the measurement of triangles. When it treats of plane 
triangles, it is called Plane Trigonometry; and as the engineer 
will continually meet in his studies of higher mathematics the 
terms used m plane trigonometry, it is advantageous for him to 
become familiar with some of the principles and definitions 
relating to this branch of mathematics. 

The circumferences of all circles contain the same number of 
degrees, but the greater the radius the greater is the absolute 
measures of a degree. The circumference of a fly wheel or the 
circumference of the earth have the same number of degrees; 
yet the same number of degrees in each and every circumfer- 
ence is the measure of precisely the same angle. 

The circumference of a circle is supposed to be divided into 
360 degrees or divisions, and as the total angularity about the 
center is equal to four right angles, each right angle contains 
90 degrees, or 90°, and half a right angle contains 45°. Each 
degree is divided into 60 minutes, or 60'; andj for the sake of 
still further minuteness of measurement, each minute is divided 
into 60 seconds, or 60". In a whole circle there are, therefore, 
360 X 60 X 60=1,296,000 seconds. The annexed diagram, Fig. 
67, exemplifies the relative positions of the 

Sine, Tangent, 

Co-sine, Co-Targent, 

Versed Sine, Secant, and 

Co-secant 
of an angle. 



Hand Book of Calculations, 



*37 




radius 



4.v-sine, H- cosine* * 

<- rcuLus «• 

Fig. 67. 

Definitions. 

1. The Complement of an arc is 90° minus the arc. 

2. The Supplement of an arc is 180° minus the arc. 

3. The Sine of an angle, or of an arc, is a line drawn from 
one end of an arc, perpend icular to a diameter drawn through 
the other end. 

4. The Cosine of an arc is the perpendicular distance from 
the center of the circle to the sine of the arc; or, it is the same 
in magnitude as the sine of the complement of the arc. 

5. The Tangent of an arc is a line touching the circle in one 
extremity of the arc, and continued from thence, to meet a 
line drawn through the center and the other extremity. 

6. The Cotangent of an arc is the tangent of the complement 
of the arc. 

R i mark. — The Co is but a contraction of the word complement. 

7. The Seen nt of an arc is a line drawn from the center of 
the circle to the extremity of the tangent. 

8. The Cosecant of an arc is the secant of the complement. 

9. The Versed Sine of an arc is the distance from the ex- 
tremity of the arc to the foot of the sine. 

For the Hake of brevity, these technical terms are contracted 
thus: for sine A B, we write sin. AB; for cosine AB, we write 
cos. AB; for tangent AB, we write tan. A B, etc. 



'38 



Hand Book of Calculations. 



GEOMETRICAL PROBLEMS. 



Fig. 68. 




The following problems are to be solved 
by the use of the dividers and rule : 

Problem I. To bisect (cut in two) a 
straight line, or an arc of a circle. Fig. 
68. From tbe ends A B as centers, 
describe arcs cutting each other at C 
and D, and draw C D, which, cuts the 
line at E or the arc at F. 



Problem II. To draw a perpendicular to a straight line, or 
a radial line to a circular arc, Fig. 68. Operate as in the 
foregoing problem. The line C D is perpendicular to A B ; 
the line CD is also radial to the arc A B. 



Problem III. To draw a per- 
pendicular to a straight line, from 
a given point in that line, Fig. 
69. With any radius from any 
given point A in the line B C, 
cut the line A B and C. Next, 1 — ! 

. « ^ A (• C 

with a longer radius describe arcs Fig. 69. 

from B and C, cutting each other at D, and draw the perpen 

dicular D A. 




Fig. 70. 



2d Method, Fig. 70. From any 
center F above B C, describe a cir- 
cle passing through the given point 
A, and cutting the given line at D; 
draw D F, and produce it to cut 
the circle at E ; and draw the per- 
pendicular A E« 



Hand Book of Calculations. 



J 39 



3d Method, Fig 71. From A describe an 
arc E C, and from E with the same radius, 
the arc A C, cutting the other at C ; through 
C draw a line E U D and set off CD equal 
to C E, and through D draw the perpen- 
dicular A D. 



JC/ 



Fig. 71. 




Fig. 72. 



Problem IV. To draiv a perpendic- 
ular to a straight line from any point 
without it, Fig. 72. From the point A 
with a sufficient radius cut the given line 
at F and G ; and from these points 
describe arcs cutting at E. Draw the 
perpendicular A E. 

Note. 



If there be no room below the line, the intersection may be 
taken above the line, that is to say, between the line and the 
given point. 



2d Method, Fig. 73. From 
any two points B C at some 
distance apart, in the given 
line, and with the radii B A, 
C A, respectively, describe arcs 
cutting at A I). Draw the 
perpendicular A I>. 



-% 



Fig. 73. 
Problem V. To draw a parallel line through a given point, 
Fig. 74-. With a radius equal to the given point from the 

given line A B, describe the 

arc D from B taken consider- 
ably distant from C. Draw 
L ^ the parallel through <7to touch 
the arc D. 



Fig 74, 



140 



Hand Book of Calculations, 



Second Method, F-g. 75. From A, the given point describe 

A/ !» 



/ 



f 



Fig. 75. 



the arc F D, cutting the given 
line at F ; from F with the 
same radius, describe the arc E 
A, and set off F D equal to E A. 
Draw the parallel through the 
points A D, 

Note. 

When a series of parallels are required perpendicular to a base 
line A B, they may be drawn as in figure 76 through points in 

the base line set off at the 
required distances apart. 
This method is convenient 
also where a succession of 
b parallels are required to a 
given line C D, for the per- 
pendicular may be drawn 
to it, and any number of 
parallels may be drawn on 
the perpendicular. 

Problem VI. To divide a line into a number of equal parts, 
Kg. 77. 

To divide the line A B into, say 5 parts. From A and B 
draw parallels A C, B D on opposite sides; set off any con- 
venient distance four times (one 
less than the given number), 
from A on A C, and on B on B 
F; join the first on A O to the 
fourth on B D, and so on. The 
lines so drawn divide *A B 
required. 






Fig. TO. 



as 





d\ Second Method, Fig. 78. Draw the 
* line at A C, at an angle from A, set off 
say, five equal parts; draw B 5, and 
draw parallels to it from the other 
B points of division in A C. These par- 
allels divide A B as required. 



Hand Book of Calculations. 



141 



GEOMETRICAL PROBLEMS. 

Problem VII. Upon a straight line to draw an angle equal 
to a given angle, Fig. 79. Let A be the given angle and F G 
the line. With any radius from the points A and F, describe 
arcs D E, I H % cutting the sides of the angle A and the line 
FG. 





Fig. 79. 

Set off the arc I H equal to D E and draw F H. 

/'is equal t-o A as required. 



Problem VIII. To bisect an angle, 
Fig. 80. Let A B be the angle; on 
the center C cut the sides at A B. On % 
A and B as centers describe arcs cut- 
ting at D dividing the angle into two 
equal parts. 



The angle 




Fig. 80. 



Problem IX. To find the 
center of a circle or of an arc of 
a circle. First for a circle, . % /' 
Fig. 81. Draw the chord A B, *\ 
bisect it by the perpendicu- 
lar C D, bounded both ways by 
the circle; and bisect C D for 
the center G. 




S' 



Fig. 81. 



142 



Hand Book of Calculations. 



GEOMETRICAL PROBLEMS. 

Problem X. Through two given points 
to describe an arc of a circle with a given 
radius. Fig. 82. On the points A and B 
as centers, with the given radius, describe 
arcs cutting at C; and from C, with the 
same radius, describe an arc A B as re- 
quired. 




Fig. 82. 




Second, for a circle or an arc, Fig. 83. 
Select three points A B Cm the circum- 
ference, well apart; with the same radius; 
describe arcs from these three points cut- 
ting each other, and draw two lines D E, 
F G, through their intersections accord- 
ing to Fig. 68. The point where they 
cut is the center of the circle or arc. 



Fig. 83. 



Problem XL To describe a circle passing through three 
given points, Fig. 83. Let A B C be the given points and 
proceed as in last problem to find the center 0, from which the 
circle may be described. 

Note. 

This problem is variously useful; in finding the diameter of 
a large fly wheel, or any other object of large diameter when 
ouly a part of the circumference is accessible; in striking out 
arches when the span and rise are given, etc. 

Problem XII. To draw a tangent to a circle from a given 
point in the circumference, 
Fig. 84. From A set off 
equal segments A B, A D, 
join B D and draw A E, 
parallel to it, for the tan- 
gent. 




Fig. 84. 



Ha mi Book of Calculations. 



*43 






Us 




Problem XIII. To draw 
tangents to a circle from points 
without it, Fig. 85. From A 
with the radius A C, describe 
an arc B D, and from 
with a radius equal to the di- 
ameter of the circle, cut the 
arc at B D; join B C, C D y 
cutting the circle at E F, and 
draw A E, A F, the tangents. 



*J> 



Fig. 85. 

Problem XIV. Between two inclined lines to draw a series 
of circles touching these lines and touching each other, Fig. 86. 
Bisect the inclination of the given lines A B, C D by the line 
JV" 0. From a point P in this line draw the perpendicular 
P B to the line A B, and 
on P describe the circle 
B D s touching the lines 
and cutting the center 
[meat/?. From/? draw 
E /'perpendicular to the 
center line, cutting A B 
at F, and from F de- 
scribe an arc E G, cut- 
ting A B at G. Draw G H parallel to B P, giving H, the 
center of the next circle, to be described with the radius HE, 
and so on for the next circle I N. 



Problem XV. To construct a triangle 
on a given base, the sides being given. 

First. An equilateral triangle, Fig. 87. 
On the ends cf a given base A B, with 
A B as a radius describe arcs cutting at C, 
and draw A (\ C B. 




Fig. 86. 




Pig. 87. 



j 44 



Hand Book of Calculations. 




Fig. 88. 



Second. A triangle of un- 
equal sides, Fig. 88. On either 
end of the base A D with the side 
B as a radius, describe an arc; 
and with the side G as a radius on 
the other end of the base as a A 
center describe arcs cutting the 
arc at E. Join A E, D E. B 

Note. c 

This construction may be used 
for finding the position of a point G or E at given distances 
from the ends of a base, not necessarily to form a triangle. 

Pkoblem XVI. To construct 
a square rectangle on a given 
straight line. 

First. A square, Fig. 89. On 
the ends A B as centers, with the 
line A B as radius, describe arcs 
cutting at G; on G describe arcs 
cutting the others at D E; and on 
D and E cut these at F G. 
Draw A F B G and join the in- 
Fi g . 89. tersections H L 

Second. A rectangle, Fig. 90. 
On the base E F draw the perpen- 
diculars EH, F G, equal to the 
height of the rectangle and join G 
H. 

Fig 90. 
Problem XVII. To construct a parallelogram of which the 
sides and one of the angles are given, Fig. 91. Draw the side 
D E equal to the given length A, and set off the other side D F 

equal to the other length 
B, forming the given angle 
G. From E with D F as 
radius, describe an arc, and 
from F, with the radius D E 
cut the arc at G. Draw 
F G, E G. Or, the remain- 
ing sides may be drawn as 
parallels to D E. D F. 





-** 








! H 


■ 
I 












\ 


/ 


\ 




\ 






, 


/ 


c*» 


-'"' 


B 



*Lr 










"N 


V 




'' V 




v. 


\ 




s \ 




A 


\ 




/ 




' \ 


\ E ' 






pi v 


• 


A r 








^4 


c 


T\ 



Fig. 91 



Ha?id Book of Calculations, 



W5 




GEOMETRICAL PROBLEMS. 



Problem XYIII. To (/escribe a 
circle about a triangle, Fig. 92. 

Bisect two sides A B, A C of the 
triangle at E F, and from these 
points draw perpendiculars cutting 
at K. On the center K, with the 
radius K A draw the circle .4 B C. 



Fig. 92. 
Problem XIX. To describe a circle about a square, and to 
inscribe a square in a circle. Fig. 94. 

First. To describe the circle. Draw 
the diagonals A B, C D of the square, 
cutting at E; on the center E with the 
radius E A describe the circle. 

Second. To inscribe the square. 
Draw the two diameters A B, G D at 
right angles and join the points A B, 
C D to form the square. 

Note. 

In the same way a circle may be described about a triangle. 
Problem XX. To inscribe a circle on a square, and to 
describe a square about << circle, Fig. 94. 

First. To inscribe the circle. Draw 
the diagonals A B, CD of the square, cut- 
ting at E; draw the perpendicular E Fto 
one side, and witn the radius E Z 7 describe 
the circle. 

Second. To describe the square. Draw 

two diameters A B, C D at right angles, 

1* and produce them; bisect the angle/) ED 

Fig. M- at the center by the diameter F G, and 

through Fasx6 G draw perpendiculars A C, B D, and join the 

points A D and B where they cut the diagonals to complete 

the square. 





i^6 



Hand Book of Calculations. 




Problem XXI. To inscribe a 
circle in a triangle, Fig. 95. Bi- 
sect two of the angles A G of the 
triangle by lines cutting at D; 
from D draw a perpendicular D 
E to any side, and with D E as 
radius describe a circle. 



Problem XXII. To inscribe a 

pentagon in a circle, Fig 96. Draw 
two diameters A G, B D at right 
angles cutting at 0; bisect A 
at E, and from E with radius E B 
cut the circumference at G H 
and with the same radius step 
round the circle to / and K; join 
the points to form the pentagon. 







B 










k^ 


^7- 


n^ 






' 




/ 




\ 




*l\ 


; 









1 / / \ 


A.r 










f / r 




rc^ 








\ 




Fig~97. 

Problem XXIY. To inscribe a 
hexagon in a circle, Fig. 98. Draw 
a diameter A G B; from A and B as 
centers with the radius of the circle 
A G, cut the circumference at D, 
E, F, G, and draw A D, D E, etc., 
to form the hexagon. 
Note. 

The points D E, etc., may be 
found by stepping the radius (with the 
dividers) six times round the circle. 



D 

Fig. 96 
Problem XXIII. To construct 
a hexagon upon a given straight 
line, Fig. 97. From A and B the 
ends of the given line describe arcs 
j D cutting at G; from G with the 
radius G A describe a circle. With 
the same radius set oh* the arcs 
AG, G Fund BD, DE. Join the 
points so found to form the hexa- 
gon. 




Fig. 98. 



Hand Book of Calculations. 



147 



Problem XXV. To describe an octagon on a given straight 



line, Fig. G9. 




Produce the given line A B both ways and 
draw perpendiculars ^4 E, B F; 
bisect the external angles A and 
B by the lines A H, B C, which 
make equal to A B. Draw C D 
and H G parallel to A E and 
equal to A B; from the center 
G D, with the radius A B, cut 
the perpendiculars at E F, and 
draw E F to complete the hexa- 
gon. 



A B 

Fig. 99. 

Problem XXYJ. To convert a 
square into an octagon, Fig. 100. 

Draw the diagonals of the square 
cutting at e; from the corners A B C 
D, with A e as radius, describe arcs 
cutting the sides at g, h, etc.; and 
join the points so found to complete 
the octairon. 





m i 

Fig. 100. 



Problem XXVII. To inscribe 
an octagon in a circle, Fig. 101. 

Draw two diameters A C, B D, 
at right angles; bisect the arcs A 
B, B C, and C at e, f, etc. , to form 
the octagon. 



d Fig. 101. 

Problem XXVIII. To describe 
an octagon about a circle, Fig. 102. 

Describe a square about the given 
circle A B, draw perpendiculars h, 
k and 6' to the diagonals, touching 
the circle, to form the octagon. Or, 
the points h, h, etc., may be found 
by cutting the sides from the corners. 



R t 



Fig. 102. 



14S 



Hand Book of Calculations. 




GEOMETRICAL PROBLEMS. 

Problem XXIX. To describe an ellipse wlien the length and 
breadth are given, Fig. 103. On the center G, with A E as 
radius, cut the axis A B at 
F and G, the foci; fix a 
couple of pins into the axis 
at F and 67, and loop on a 
thread cr cord upon them 
equal in length to the axi 
A B, so as when stretched t.> 
reach the extremity of the 
conjugate axis, as shown in 
dot-lining. Place a pencil 
or drawpcint inside the cord, 
as at H, and guiding the pencil in this way, keeping the cord 
equally in tension, carry the pencil round the pins F, G, and 
so describe the ellipse. 

Note. 

The ellipse is an oval figure, like a circle in perspective. 
The line that divides it equally in the direction of its great 
length is the transverse axis, and the line which divides the 
opposite way is the conjugate axis. 

Second Method. Along the straight edge of a piece of stiff 
paper mark off a distance a c equal to A C, half the transverse 

axis; and from the same 
point a distance a b equal 
to C D, half the conju- 
gate axis. Place the slip 
so as to bring the point b 
on the line A B of the 
transverse axis, and the 
point c on the line D E; 
Fig. 104. and set off on the drawing 

the position of the point a. Shifting the slip, so that the point 
b travels on the transverse axis, and the point c on the conju- 
gate axis, any number of points in the curve may be found, 
through which the curve nay be traced. 




Hand Book of Calculations. /./y 



PROPORTION, OR THE RULE OF THREE. 

The Bide of Three, or proportion, is one of the most useful 
in the whole range of mathematics; a rule by which, when 
three numbers are given, a fourth number is found, which 
hears the same relation to the third as the second does to the 
first ; or a fourth number is found bearing the same relation 
to the third as the first does to the second. 

Proportion is the relation of one quantity to another. This 
relation may be expressed either by the difference of the 
quantities or by their quotient. In the former case it is called 
arithmetical relation, in the latter geometrical proportion or 
simple proportion. 

Proportion differs from ratio. Ratio is properly the rela- 
tion of two magnitudes or quantities of one and the same 
kind ; as the relation of 5 to 10 or 8 to 16. Proportion is the 
sameness or likeness of two such relations ; thus 5 is to TO, as 
8 to 16, or, A is to B as C is to D ; that is, 5 bears the same 
relation to 10 as 8 does to 16. Hence we say such numbers 
are in proportion. 

A proportion is an equality of ratios, and as ratio is the 
measure of the relations of two like quantities, 

It is determined by dividing the first quantity by the second. 
Thus: 

The ratio of 6 to 3 is 2, or of $8 to $2 is 4. 
8 : 2 = 16 : 4, is a proportion. 

The equality is generally indicated by writing :: between the 
ratios, thus: 

8 : 2 :: 16 : 4 indicates a proportion and is read, eight is to 
two, as, sixteen is to four. 

Note. 

The Bign : is an abbreviated form of -j- and has a like mean- 
ing. 

In proportion, three quantities are given, the problem being 
to rind the fourth, as 2 is to 4 as 6 is to what number — expressed 
thus: 2 :1::(J: ? 

Now then: multiply the second term by the third term and 
divide this product by the first term. 

_ 4x6 = 24. 
24-f-2=12, which is the required number. 



150 Hand Book of Calculations. 



Rule. 

Of the three given numbers, place that for the third term 
which is of the same kind with the answer sought. 

Then consider, from the nature of the question, whether the 
answer will be greater 01 less than this term. If the answer is 
to be greater, place the greater of the two numbers for the sec- 
ond term, and the less number for the first term; but if it is to 
be less, place the less of the two remaining numbers for the 
second term, and the greater for the first; and in either case 
multiply the second and third terms together, and divide the 
product by the first for the answer,, which will always be of the 
same denomination as the third term. 

Note, 

If the first and second terms contain different denominations, 
they must both be reduced to the same denomination; and 
compound numbers to integers of the lowest denomination 
contained in it. 

Example. 

2. If 40 tons of iron cost , what will 130 tons cost ? 

Tons. Dor -ons. 

40 :4:0:.130 
130 

13500 
450 



40)5850|0 



1462.5 dollars Ans. 

The Terms of a ratio are the two numbers compared. The 
Antecedent is the first term of a ratio, the Consequent is the 
second term, and the two terms together are called a Couplet. 
An Inverse Ratio is the ratio formed by inverting the terms of 
a given ratio. Thus 8:9 is the inverse of 9:8. 

Each term of a proportion is called a Proportional ; the first 
and fourth terms are called Extremes ; and the second and 
third term, Means. When the two means are the same num- 
ber, that number is a Mean Proportional between the two 
extremes. 



Hand Book of Calculations. l 5 l 



THERMO-DYNAMICS. 



Heat is treated generally in scientific books under the head- 
ing of tliermo-clynamics. 

This term is made from two Greek words which signify, 
lectively heat-power; i. e., the power which is produced by 
the combustion or burning of fuel. 

Without heat there would be no steam engine or steam boiler, 
and no engineer nor fireman; hence, the consideration of its 
nature and management and the calculations connected with 
its employment stand first in the order of subjects, heat, water, 
steam, now to be explained in their relations to mathematical 
calculations. 

HEAT. 

When two bodies in the neighborhood of each other have un- 
equal temperatures, there exists between them a transfer of heat 
from the hotter of the two to the other. 

The tendency towards an equalization, or towards an equilib- 
rium of temperatures is universal, and the passage of heat takes 
place in three ways: 

1. By radiation. 

2. By conduction. 

3. And by convection, or carriage from 
one place to another by heated currents. 

RADIATION OF HEAT. 

Radiant heat traverses air without heating it. 

By means of a simple apparatus it has been ascertained that 
the proportion of the total he^>t radiated from different combus- 
tibles are as follows : 

Radiated heat from wood, ..... nearly £ 
<lo do wood charcoal " \ 

do do oil, " * 

These values serve to show that radiation from heat is con- 
siderable; and that flameless carbon such as is wood charcoal, 
radiates mere than oil, which is also nearly pure carbon, does 
with its more brilliant consumption. 



1 52 Hand Book of Calculations, 



THERMODYNAMICS. 

The heat which is experienced by holding the hand near the 
flame of a candle, by its side, is the heat caused by radiation, 
while the heat felt by the hand held over the flame, is the heat 
conveyed by convection. But it is to be noticed that while the 
radiant heat is dissipated all round the flame, the diameter of 
the upward current is little more than that of the flame, and 
the conveyed heat is therefore concentrated in a narrow compass. 

With respect to the heated bodies, apart from combustibles 
as such, the radiation or throwing out of heat implies the oppo- 
site process of absorption; and the radiators are likewise the 
best absorbents of heat. All bodies possess the property of 
radiating heat. The heat rays proceed in straight lines, and 
the intensity of the heat radiated from any one source of heat 
becomes less as the distance from the source of heat increases. 

This decrease is governed by a great natural law, which is this: 
the intensity decreases in the inverse ^atio of the square of the 
distance; that is to say, for example, that at any given distance 
from the source of radiation, the intensity of the radiant heat 
is four times as great as it is at twice the distance. 

When a polished body like sheet tin, steel or silver is struck 
by a ray of light it absorbs a part of the heat and reflects the 
rest. The greater or less proportion of heat absorbed by the 
body is the measure of its absorbing poiver, and the reflected 
heat is the measure of its reflecting poiver. 

The reflecting power of a body is the complement of its 
absorbing power; that is to say, that the sum of the absorbing and 
reflecting powers of all bodies is the same, which amounts to 
this, that a ray of heat striking a body is disposed of by absorp- 
tion and reflection together, that which is not absorbed being 
naturally reflected. 

CONDUCTION OF HEAT. 

Conduction is the movement of heat through substances, or 
from one substance to another m contact with it. A body 
which conducts heat well is called a good conductor of heat; if 
it conducts heat slowly it is a bad conductor of heat. Bodies 
which are finely fibrous, as cotton, wool, wadding, finely divid- 



Hand Book of Calculations. 



53 



THERMODYNAMICS. 

ed charcoal, are the worst conductors of heat. Liquids and 
gases are had conductors; but if suitable provision is made for 
the free circulation of fluids they may abstract heat very quickly 
by contact with heated surfaces acting hy convection. The 
table contains the relative conducting powers of metals and 
earth according to M. Despretz. 

Relative Lnterxal Conducting Power of Metals. 



Substance. 


Relative Conduct- 
ing Powers. 


Substance. 


Relative Conduct- 
ing Powers. 


Gold 

Platinum 


1000 

981 


•Zinc 


363 


Tin 


..... 304 


Silver 


.... 973 


iLead 


180 


Copper 

Brass 


892 

... 749 


! Marble 


24 


Porcelain 


12 


Cast Iron 


502 


Terra Cotta . . . 


11 


Wrought Iron . 


374 







CONVECTION OF HEAT. 
Convected or carried heat is that which is transferred from 
one- place to another by a current of liquid or gas; for example, 
by the products of combustion in a furnace towards the heating 
surface m the flues of a boiler. 

THE MECHANICAL THEORY OF HEAT. 
Heat and mechan- 
ical force are identi- 
cal and convertible. 
Independently of the 
Tried in in through 
which heat may be 
developed into me- 
chanica] action the 
tame quantity of hen t 
it ft bo1v< d into ilir 
tame total quantity 
of work. 




1 54 Hand Book of Calculations. 

THERMO-DYNAMICS. 

The unit of heat is that which is required to raise 1 lh. of 
water, at 39 degrees Fahr., 1 degree. If 2 lbs. of water be 
raised 1 degree or 1 lb. be raised 2 degrees in temperature, the 
expenditure of heat is the same in amount, namely, two degrees 
of heat, and to express the mechanical equivalent of heat the 
comparison lies between the unit of heat on the one part and 
the unit of work — a foot pound — on the other. 

The most precise determination yet made of the numerical 
relation subsisting between heat and mechanical work was 
obtained by the following experiment by Dr. Jorles: He con- 
structed an agitator, Fig. 105 consisting of a vertical shaft 
carrying a brass paddlewheel, of which the paddles revolved 
between stationary vanes, which served to prevent the liquid 
in the vessel from being bodily whirled in the direction of rota- 
tion. The vessel was filled with water and the agitator made 
to revolve by means of a cord wound round the upper part of 
the shaft and attached to a weight which descended m front of 
a scale by which the work done was measured. When all cor- 
rections had been applied, it was found that the heat communi- 
cated to the water by the agitation amounted to one pound 
degree Fahrenheit for every 772 foot pounds of work expended 
in producing it. Hence it was deduced that one unit of heat 
was capable of raising 772 lbs. weight 1 foot in height. The 
mechanical equivalent of heat, known as Jorles'' equivalent, is 
772 foot lbs. for 1 unit of heat. Sperm oil was also tried as the 
fluid medium and it yielded the same result as water. 

According to the mechanical theory of heat, in its general 
form, heat, mechanical force, electricity, chemical affinity, 
light and sound are but different manifestations of motion — 
thus, the intense heat of the furnace is the result of an amazing 
rapidity of motion taking place among the particles during 
the decomposition of the mass of fuel. 



Hand Book of Calculations. 



r 55 



n 



J£) 



THERMOMETERS. 

The action of Thermometer is based on the change of volume 
to which bodies are subject with a change of temperature, and 
they serve, as their name implies, to measure temperature. 
Thermometers are filled with air, water, or mercury. Mercurial 
thermometers are the most convenient, because the 
most compact. They consist of a stem or tube of 
glass, formed with a bulbous expansion at the foot 
to contain the mercury, which expands into the tube. 
The stem being uniform in bore, and the apparent 
expansion of mercury in the tube being equal for 
equal increments of temperature, it follows that if 
the scale be graduated with equal intervals, these 
will indicate equal increments of temperature. A 
sufficient quantity of mercury having been intro- 
duced, it is boiled to expel air and moisture, and the 
tube is hermetically sealed. The freezing and the 
boiling points on the scale are then determined 
respectively by immersing the thermometer in melt- 
ing ice and afterwards in the steam of water boiling 
under the mean atmospheric pressure, 14.7 lbs. per 
square inch, and marking the two heights of the 
column of mercury in the tube. The interval 
between these two points is divided into 180 degrees 
for Fahrenheit's scale, or 100 degrees for the Centi- Fig. 106. 
grade scale, and degrees of the same interval are continued 
above and below the standard points as far as may be necessary. 
It is to be noted that any inequalities in the bore of the glass 
must be allowed for by an adaptation of the lengths of the grad- 
uations. The rate of expansion of mercury is not strictly 
constant, but increases with the temperature, though, as already 
referred to, this irregularity is more or less nearly compensated 
by the varying rates of expansion of glass. 

In the Fahrenheit Thermometer, used in Britain and Ameri- 
ca, the number 0° on the scale corresponds to the greatest 
degree of cold that could be artificially produced when the 
thermometer was originally introduced. 32° ("the freezing- 
point v ) corresponds to the temperature of melting ice, and 212° 



(%;*) 



i 



'■56 



Hand Book of Calculations. 



THERMOMETERS. 

to the temperature of pure boiling water — in both cases under 
the ordinary atmospheric pressure of 14.7 lbs. per square inch. 
Each division of the thermometer represents 1° Fahrenheit, 
and between 32° and 212° there are 180°. 



THE MEASUKEMENT OF HEAT. 

Temperature means the sensible beat in anything, and is 
measured by the Thermometer. 

There are three kinds of Thermometers in general use — 
Fahrenheit's, Centigrade, and Reaumur's. 





F 


c 




R 




BOILING 












A 


212° 


s\ 


109° 


/> 


80" 


180 




100 




80 




FREEZING >' 




\r 





w 


O 




32° 










- 












< 




( 


> 


< 


s 



Fig. 107. 

The on these scales is called Zero, all above the is plus, 
while all below is minus; thus a temperature of 10° below Zero 
is written —10° 



Hand Book of Calculations. 1 57 

THE MEASUREMENT OF HEAT. 

In Fahrenheit's, the space between Freezing and Boiling 
points is divided into 180 d3grecs, Freez- 
ing being 32° and Boiling 212°. 

In Centigrade, Freezing is and Boiling 100°, the space 
being thus divided into 100 parts; hence 
its name. 

In Reaumur's, Freezing is and Boiling is 80°, the space 
bemg thus divided into 80 parts. 

In the above diagram it is readily seen that 
180° F — 100° C — 80° E; 

from which we can get the rules for comparing degrees of 
temperature on one scale with the degrees on. another. 

In the Cent. grade Thermometer, used in France and in most 
other countries in Europe, C corresponds to melting ice, and 
100° to boiling water. From the freezing to the boiling point 
there are 100 c 

In the Reaumur Thermometer, used in Russia, Sweden, Tur- 
key, and Egypt, 0° corresponds to melting ice, and 80° to boil- 
ing water. From the freezing to the boiling point there are 80°. 

Centigrade temperatures are converted into Fahrenheit tem- 
peratures by multiplying (he former by 9 and dividing by 5, 
and adding 32° to the quotient; and conversely, Fahrenheit 
temperatures are converted into Centigrade by deducting 32° 
and taking Hhs of the remainder. 

Reaumur degrees are multiplied by § to convert them into the 
equivalent Centigrade degrees; conversely, libs cf the number 
of Centigrade degrees give their equivalent in Reaumur degrees. 

Fahrenheit is converted into Reaumur by deducting 32° and 
taking l ths of the remainder, and Reaumur into Fahrenheit by 
multiplying by J, and adding 32° to the product. 

PYROMETERS. 

Pyrometers are employed to measure temperatureH above the 
Innliug point of mercury, about 676° F. 



I $8 Hand Book of Calculations. 



THERMODYNAMICS. 



PYROMETEK. 



Wedgwood's pyrometer, invented in 1782, was founded on 
the property possessed by clay of contracting at high tempera- 
tures. The apparatus consists of a metallic groove, 24 inches 
long, the sides of which converge, being half an inch wide 
above and three-tenths below. The clay is made up into little 
cylinders or truncated cones, which fit the commencement of 
the groove after having been heated to low redness; their sub- 
sequent contraction by heat is determined by allowing them to 
slide from the top of the groove downwards till they arrive at a 
part of it through which they cannot pass. 

In Darnell's pyrometer the temperature is measured by the 
expansion of a metal bar inclosed in a black-lead earthenware 
case, which is drilled out longitudinally to T 3 o inch in diameter 
and 7k inches deep. A bar of platinum or soft iron, a little 
less in diameter, and an inch shorter than the bore, is placed 
in it and surmounted by a porcelain index 1 J inches long, kept 
in its place by a strap of platinum and an earthenware wedge. 
When the instrument is heated, the bar, by its greater rate of 
expansion compared with the black-lead, presses forward the 
index, which is kept in its new situation by the strap and wedge 
until the instrument cools, when the observation can be taken 
by means of a scale. 

Another means of estimation, based on the melting points of 
metals and metallic alloys, is applied simply by suspending in 
the heated medium a piece of metal or alloy of which the melt- 
ing point is known, and if necessary, two or more pieces of 
different melting points, so as to ascertain, according to the 
pieces which are melted and those which continue in the solid 
state, within certain limits of temperature, the heat of the fur- 
nace. A list of melting points of metals and metallic alloys is 
given in a subsequent chapter. 



Hand Book of Calculations. i $g 



THERMODYNAMICS. 



LUMINOSITY AT HIGH TEMPERATURES. 



The luminosity or shades of temperature have been observed 
by M. Pouillet by means of an air-pyrometer to be as follows: — 

Shade. Temperature. 

Fahrenheit. 

Nascent Red. 977° 

Dark Red 1292 

Nascent Cherry Red 1472 

( Jherry Red 1652 

Bright Cherry Red 1832 

Very Deep Orange 2012 

Bright Orange 2192 

White 2372 

u Sweating" White. . 2552 

Dazzling White 2732 

A bright bar of iron, slowly heated in contact with air, 
assumes the following tints at annexed temperatures (Claudel): 

Fahrenheit. 

1 . Cold iron at about 54° 

•l. Yellow at 437 

3. Orange at 473 

4. Red at 509 

5. Violet at 531 * 

6. Indigo at 550 

7. Blue at 559 

8. Green at 630 

9. Oxide Gray (gris d'oxyde) at 753 



l6o Maud Book of Calculations. 



HORSE POWER. 



A horse power is merely an expression for a certain amount 
of work and involves three elements — 
1 . Force. 

2. Space; and 
3. Time. 

If the force be expressed in pounds, and the space passed 
through in feet, then we have a solution of and the meaning 
for, the term foot-pound ; from which it will be seen that a 
foot-pound is a resistance equal to one pound moved upwards 
one foot. The work done in lifting thirty pounds through a 
height of fifty feet is fifteen hundred foot-pounds. 

Now if the foot-pounds required to do a certain amount of 
work involve a specified amount of time during which the work 
is performed and if this number of foot-pounds is divided by 
the equivalent number representing one horse power (which 
number will be dependent upon the time) then the resulting 
number will be the horse power developed. 

Example. 

Suppose the 1500 foot-pounds just spoken of to have acted 
in one second. To find the horse power divide by 550, and the 
result will be the horse power. 

A horse power is 33,000 foot-pounds, or, in other words 33,000 
pounds lifted one foot in one minute, or one pound lifted 3:?,000 
feet in one minute, or 550 lbs. lifted one foot in one second. 

HORSE POWER OF THE STEAM ENGINE. 

The capacity for work of a steam engine is expressed m the 
number of horse powers it is capable of developing. 



Hand Book of Calculations. 161 



HORSE POWER. 

There are three kinds of horse power spoken and written 
about which engineers should learn to distinguish — these are 
1. Xominal, 

2. Indicated, and 

3. Effective. 

Engineers and others who have not carefully considered the 
matter, often use the above as synonymous — or having the same 
meaning; but in this they are wrong, as the meaning is very far 
from the same. 

Nominal horse power is an expression which is gradually going 
out of use, and is merely a convenient mode of describing the 
dimensions of a steam engine for convenience of makers and 
purchasers of steam engines. 

Indicated horse power is the true measure of the work done 
within the cylinder of the steam engine and is based upon no 
assumptions, but is actually calculated. The things necessary 
to be known m order to make the figures are: 

1. The diameter of the cylinder in inches. 

2. Length of stroke m feet. 

3. The mean effective pressure — that is, the average pressure 
of the steam on the piston during the full length of the stroke; 
and 

4. The number of revolutions per minute. 

Effective horse power is the amount of work which an engine 
is capable of performing, and is the difference between the 
indicated horse power and horse power required to drive the 
engine when it is running unloaded. 

Note. 
Engine rating, guarantees, etc., are usually based upon tbe 
indicated horse power, owing to the ease and accuracy with 
which it can be determined. 

Rule for calculating horse power. 

1. Find the area of the piston. 

2. Find the j)ressure in lbs. on the piston, by yrraltiplying 
the area by the pressure per square inch. 



r 62 



Hand Book of Calculations. 



HORSE POWER. 

3. Find the space in feet travelled through by the piston per 
minute, by multiplying the length of stroke in feet by twice 
the revolutions per minute. 

4. Find the foot-pounds done by the engine per minute, by 
multiplying the pressure in lbs. (2) by the travel in feet (3). 

5. Find the H. P. by dividing the foot-lbs. (4) by 33000. 

Example. 

What is the horse power of an engine, the diameter of the 
cylinder being 16 inches, length of stroke 24 inches, revolu- 
tions per minute 120, and the average pressure of steam per 
square inch on the piston 45 lbs. ? 
ft. in. 
Diam. 1 4 .7854 stroke 2 feet. 

12 256 No. of strokes 240 



16 inches. 


47124 


480 travel of 


16 


39270 


piston in 


— 


15708 


feet* 


96 

.6 






201.0624=area. 






45 lbs. pressure. 





Diam. 256 squared. 



10053120 
8042496 



9047. 8080 ^pressure on piston in lbs* 

9047.8 lbs. 
480 feet. 



7238240 
361912 



33000 



( 3)43429440 foot pounds. 



11)14476480 



131.6044 



An& 131 T 6 T horse power. 



Hand Book of Calculations. 163 

HORSE POWER. 

Second Rule. 

Instead of putting the work down step by step, it is more 
readily worked as follows: (1) Square the diameter of the piston, 
(2) multiply it by the length of stroke in feet, (3) by -twice the 
revolutions, (4) by the pressure per square inch, (5) and by 
.0000238. 

Example. 

2. What is the horse power of an engine, the diameter of 
cylinder being 13 inches, length of stroke 12 inches, revolutions 
per minute 300 and the average pressure per square inch on 
the piston 67 lbs.? 

13 diameter in inches. 
13 

39 
13 



169 

1 lensrth of stroke in feet. 



169 . 

600 twice the revolutions, or number of strokes. 



101400 

67 lbs. pressure per square inch. 



TO9800 
608400 



6793800 

238 constant multiplier. 



54350400 
S14 
13581 



161.0024400 horse power. Ans. Jfiltfj TT. P. 



J 6^. Hand Book of Calcinations* 

HORSE POWER. 

Note. 

This rule is the same as the first one except that a constant 
multiplier is used. This is found by dividing .T854 by 33000, 
which equals .0000238. This very considerably shortens the 
calculation as will be observed by comparing the two examples 
given under the rules, i. e., the 16"x24 // and the 13"xl2" engine. 
Example for Peactice. 

3. What is the horse power of an engine, the diameter of 
cylinder being 24 inches, length of stroke CO inches, revolu- 
tions per minute GO, and the average pressure being 43to 3 o? 

Ans. : 

4. What is the horse power of an engine, the diameter of 
cylinder being 6 inches, length of stroke 9 inches, revolutions 
per minute 400, and the average pressure of steam on the piston 
being 45 lbs. ? Ans. : 

5. What is the horse power of a pair of engines, the diameter 
of the cylinder being 12 inches, length of stroke 30 inches, rev- 
olutions per minute 90, and the average pressure of steam 38 
lbs. ? Ans. : 

IMPORTANT. 

In the rules given for estimating the power of different forms 
of the steam engine, it will be observed that the area of the piston 
is a quantity winch is known to a close fraction; the piston 
speed infect per minute is assumed to be correct, and the reduc- 
tion of the foot-pounds to horsepower by dividing by 33,000 is 
the same in all rules; but the average pressure of steam is the 
doubtful part of the calculation. 

The reasons f r this are various, such as the varied expan- 
sion, wire-drawing, steam working against itself in the cylin- 
der, condensation, cramping of the exhaust, etc., etc. These 
defects, as well as the average pressure of the steam {and the 
combined pressure of the steam and the vacuum) are clearly 
shown by the Indicator; hence, while the rules given are suffi- 
ciently close for every -day practice, it is important to bear in 
mind that all questions of the power of engines are much more 
accurately determined by the Indicator. 



Hand Book of Calculations. 165 



HORSE POWER. 

Third Rule. 
To compute horse power of engines by a short rule process. 

Rule. 

Multiply the diameter of the cylinder (in inches) by itseK 
and by the distance in feet travelled by the piston per mmute 
and divide by 42,000. The quotient gives the horse power for 
each lb. of mean effective pressure. 

Example. 

6. What is the horse power of a pair of 24x24 horizontal high 
pressure engines with 120 revolutions per minute. 

24x24 = 576 
2X120= 240 



23040 
1152 

42,000)138240(3.291 

126 2 for 2 cylinders. 



122 6.582 
84 



384 

378 



60 
42 

Ans. 6.582 for each lb. of mean effective pressure. To 
obtain total horse power, multiply by the number of lbs. of 
steam. 

Fourth Rule. 

. To compute horse power of engine, by mean effective press- 
ure, as shown by indicator. 



i66 Hand Book of Calculations. 

HORSE POWER. 

Rule. 

Multiply the area of the piston, by the mean effective press- 
ure per square inch, by the stroke in feet, by the number of 
strokes per minute (out and back being two strokes), and 
divide by 33,000. The quotient is the horse power. 

Example. 

7. What is the horse power of an engine, 9 inch bore of cylin- 
der, 20 inch stroke, and 60 revolutions per minute, with mean 
effective pressure of 42 lbs. 
Area of 9 inch per table = 636 
M. E. P. 42 



1272 
2544 


33 


,000)534240(16^ H. P. 
33 


26712 
Stroke 20" 




204 
198 


12)534240 


► 


62 


44540 J 
revolutions 120 







534240 

Rule for calculating Horse Power oe Condensing 

Engines. 

Proceed as in Rules and Examples on pages 161 and 162, add- 
ing 10 lbs. to the average pressure on the piston for the increased 
efficiency — above friction, etc.- for approximate advantage, 
gained by the use of the condenser. 

Note. 

Engine builders add from tV to \ to the nominal horse power 
of their engines as an approximation of the increased efficiency 
of their high pressure engines when fitted with condensing 
apparatus. 



Hand Book of Calculations. i6j 

HORSE POWER. 

Rule for finding the Horse Power of the Compound 

Engine. 
Find the horse power of each cylinder separately, then add 
the two powers together; or in other words treat the two cylin- 
ders as you would two separate engines. 

Note. 
In a compound engine a second cylinder of three or four 
times the piston area is added, called the low pressure cylinder, 
into which the exhaust steam of the first or high pressure cyl- 
inder, instead of being thrown away, is passed and made to 
yield a further amount of* work. The additional work thus 
obtained is roughly proportional to the mean effective pressure 
in the low pressure cylinder multiplied by the difference m 
area of the two pistons. By this means the power of the engine 
is increased, and the steam, when finally exhausted, is at a 
pressure so low that little or no unused work remains in it. 

Example. 
8. What is the H. P. of a compound engine whose high press- 
are cylinder is 16 inches in diameter, and low pressure 27 
inches, with 16 inch stroke, and 250 revolutions per minute. 
Estimate the mean effective pressure as 70 lbs. for the 16 inch 
cylinder and 12 lbs. for the 27 inch cylinder. Now: 
16" area — 201 27" area = 572.5 inches. 

70 pressure 12 lbs. 



14070 6870.0 

feet traverse 666 feet traverse 666 



84420 41220 

84420 41220 

84420 41220 



33,000)9370620(284 nearly. 33,000)4575420(139 nearly. 
Add 284 high pressure cylinder, non-condensing. 
139 low " " " 

423 total (newly). 



1 68 Hand Book of Calculations. 

HORSE POWER OF THE COMPOUND ENGINE. 

Example. 

9. What is the horse power of a compound engine, diameter of 
high pressure cylinder 27-J inches, and mean effective pressure 
throughout the stroke 36.95 lbs. per square inch. Diameter of 
low pressure cylinder 48 in., and mean effective pressure 7.35' 
lbs. per square inch, length of stroke 2 feet 6 inches, and revo- 
lutions per minute 75 ? 

Find the H. P. of each cylinder separately, then add the two- 
powers together. 

H. P. of high pressure=249.395 &c. 
do. low do. =151.139 &c. 



Combined H. P. =400.534 

POWER OF THE LOCOMOTIVE. 

The power of the locomotive is measured at the point where* 
the wheel touches the rail, and is equal to the load the locomo- 
tive could lift out of a pit by means of a rope passed over a 
pulley, and attached to the outside of the tire of one of the 
driving wheels. 

The term horse power is not generally used in speaking of 
the locomotive, as the difference in the work between it and the 
stationary engine is so great. The power of the locomotive 
resides in two places, first, the adhesive power which is derived 
from the weight on the driving wheels, and their friction and 
adhesion on the rails — it being remembered that the adhesion 
varies with the weight on the drivers and the state of the rail. 
Second, the tractive power of the locomotive, which is that 
derived from the pressure of the steam on the piston applied 
to the cranks and revolving wheels. 

Rule to find the House Power of a Locomotive. 

Multiply the area of the piston in square inches by 2 (there 
being 2 cylinders to each engine) also by two-thirds the boiler 
pressure as shown by the gauge; also by the number of revo- 
lutions per minute; also by the feet traversed by the piston- 
Divide by 33,000 and the amount will be the horse power. 



Hand Book of Calculations. i6cy 

HORSE POWER OF THE LOCOMOTIVE. 

Example. 
10. The locomotive " A. G. Darwin " has 19 inch cylinders, 
24 inch stroke, driving wheels 68 inches in diameter. It makes 
(with ease) 60 miles per hour, with boiler pressure 150 lbs. per 

square inch. 

Area of piston 283.5x2 = 567 sq. inches, 
f boiler pressure = 100 lbs. 



* To find revolutions per minute. 56700 

1 mile in 1 min. = 5280 ft. = 68360 300 revolutions. 

divide by rim of driving wheel 68" 

diam. =213.6 inches. Now: 17010000 
213.6-^-63.360 = 300 nearly. 4 



33,000)68040000 



2062 horse power. 
Note. 
This engine weighs 120,000 lbs., of which. 72,000 are on the 
drivers. By actual count it carried its own immense weight 
added to that of 8 heavy cars a mile in 47 seconds and several 
miles in 55, 58 and 60 seconds. 

Example. 
11. What isth j power of a locomotive with cylinders 19 inches 
bore, 30 inch stroke, diameter of drivers 72 inches, running 
speed 40 miles per hour, boiler pressure 160 lbs. per square 
inch. 

Area of piston 19 inches (per table page 116) = 283.5 inches. 
Steam pressure, say in this example, six-tenths of 160 = 96 
lbs. 

Now tlien, per Rule: 

283 ' 5X9 — ° X5X12 -I359f nominal H. P. 
33000 3 

Note. 

This must not be taken for the locomotive power, for it is not* 

This is the power which the engine would develope if the tires 

on the drivers were as gear wheels fitted to cogs on the track, 

so that they could not slip, and if the boiler could supply the 

■team. 



I jo Hand Book of Calculations. 

HORSE POWER OF THE STEAM FIRE ENGINE. 

Hence the rule given is merely approximate, as nothing can 
.be told about the internal workings of the engines without 
a test carefully performed with the indicator — and the results 
of the latter are modified by the tractive or adhesive power. 

Note. 

Colb urn's Rule for Calculating Power of Locomotives takes 
the full pressure of one cylinder instead of the mean average 
pressure of two. 

KlJLE FOR ELNDIKG THE HORSE POWER OF THE STEAM FlRE 

E^GIKE. 

Multiply the area of the piston by the average steam press- 
ure in pounds per square inch; multiply this product by the 
travel of the piston in feet per minute, divide this product by 
33000; seven-tenths of the quotient will be the horse power of 
the engine. 

Example. 
12. Area of piston 8 inch diameter=50.27 in. (see page 115). 
Stroke 8 inches, revolutions 150 per minute =200 feet travel. 
Average steam pressure 100 lbs. 
Now then: 50.27 area. 

100 lbs. steam. 



5027.00 

200 travel of piston in feet. 



33,000 foot-lbs.)100540000(30.4 
99 



30.4 

154 .7 

132 • 

21to 8 o horse powa*; 



Hand Book of Calculations. iji 

THE HORSE POWER OF THE STEAM BOILER. 

A great deal of trouble lias arisen from the application of 
this unit, the horse power, to the measurement of the capacity 
of steam boilers, for the boiler is only one part in the power- 
producing system. It furnishes the force. It is the magazine 
where is accumulated and stored the pressure obtained from the 
combustion of the coal. 

Xow some engines use steam much more economically than 
others, and a boiler which could furnish steam to develope 
power at the rate of 100 horses with the best of these, might not 
be able to do 40 horse power with the worst. Hence comes the 
question, what is the horse power of the boiler? 

To meet the complication which arose from this cause a 
standard of evaporation of thirty pounds of water per hour, 
from feed water of 100° Fahrenheit into steam at 70 lbs. gauge 
pressure, has been adopted as a horse power for steam boilers 
Some engines can develope a horse power on this number of 
pounds of steam per hour, others cannot, while many require more 
hence it is about the present average capacity. Both engineers 
and steam users have received this standard with unanimity, 
and so-called " boiler tests," are based upon their evaporative 
capacity, expressed in lbs. of water per hour. 

Square feet of heating surface is frequently used to express 
the horse power. This is figured from the number of square 
feet tf boiler and tube surface exposed to the action of the fire; 
but this method is not at all accurate, as the same amount of 
exposed surface will under some circumstances produce several 
times as much steam as others, but for the ordinary tubular 
boiler fifteen square feet of heating surface has been held to be 
equal to one horse power. 

The extent of the heating surface of a boiler depends on the 
length and diameter of the shell and the number and size of 
the tubes or flues. 

When setting boilers in brick work, the practice is to rack in 
the side walls to the shell a few inches belov/ the water line, 
and thus limit the heating surface. It is customary in calculat- 
ing the heating surface cf the shell, to consider that two-thirds, 
of it is exposed to the action of heat. 



n? 



Hand Book of Calculations. 



HORSE POWER OF THE STEAM BOILER. 

It is also customary to consider that the entire surface of the 
tubes or flues is exposed to the action of heat. 

From the table given below, the heating surface of any boiler 
can be obtained with ease. 



Table of Heating 


Surface of Boilers. 


Diameter of Boil- 
er Inches. 


Two thirds heating 

surface of shell per 

ft. of length. 


Diameter of 1 uhe or 
flue. Inches. 


Whole external 

heating surface per 

It. of length. 


24 


■ 4.19 


2 


.524 


26 


4.54 


*l 


.589 


28 


4.89 


H 


.655 


32 


5.59 


3 


785 


34 


5.93 


H 


.850 


36 


6.28 


H 


.916 


40 


6.98 


4 


1.05 


42 


7.33 


« 


1.18 


44 


7.68 


5 


1.31 


48 


8.38 


7 


1.83 


50 


8.73 


8 


2.09 


54 


9.42 


10 


2.62 


56 


9.77 


11 


2.88 


60 


10.47 


13 


3.40 


66 


11.52 


16 


4.19 


72 


12,57 


20 


5.24 



Rule for estimating Horse Power of Horizontal Tub- 
ular Steam Boilers. 
Mud the square feet of heating surface in the shell, heads 
and tubes, and divide by 15 for the nominal horse power. 

Example. 
What is the heating surface of a boiler having head 72 inches 
diameter, shell 18 feet long, with 100 tubes, 3 J inches in diam- 
eter ? Now then: 

12)72 inches. 



6 
6 

36 



6 feet diameter. 
.7854 
36 



47124 
23562 



28.2744 square feet of surface in one head. 



Hand Book of Calculations. iyj 



HORSE POWER OF THE STEAM BOILER. 
12)3.50000 



.29107 feet diameter of the tube. 
3.1416 



175002 

29167 
116663 
29167 
87501 

.916310472 feet circumference of 1 tube. 
18 



7330483936 
916310472 



16.4935S8656 square feet surface of 1 tube. 
1649.3589 equals square feet surface in all the tubes. 
3.1416 

6 



18.8496 circumference of shell in feet. 
18 



1507968 
188496 



339.2928 square feet of surface of shell. 
2-thirds of shell 



3)678.5856 



226.1952 

1649.358:) tubes. 



15)1875.5541 

125. horse power, neglecting the heads. 



I f 4 Hand Book of Calculations. 



ARITHMETICAL PROGRESSION. 

Arithmetical Progression is a series of numbers which 
succeed each other regularly, increasing or diminishing by a 
constant number or common difference: 

As 1, 3, 5, 7, 9, &c. ) increasing series. 
15, 12j 9, 6, 3, &c. ) decreasing series. 
The numbers which form the series are called terms. The 
f^st and the last term are called the extremes, and the others 
are called the means. 

In arithmetical progression, there are five things to be con- 
sidered, viz.-: 

1, The first term. 

2, The last term. 

3, The common difference. 

4, The number of terms. 

5, The sam of all the terms. 

These quantities are so related to each other, that when any 
three of them are given, the remaining two can be found. 

Given the first term, the common difference, and the number 
of terms, to find the last term. 

Rule. 

Multiply the number of terms, less one, by the common 
difference, and to the product add the first term. 

Example. 

What is the 20th term of the arithmetical progression, whose 
first term is one, the common difference -J? 

20-1=19 and 19x|=9|; and 9^+1=10^ Ans. 

Given the number of terms and the extremes, to find the corn-, 
mon difference. 

Rule. 

Divide the difference of the extremes by one less than the 
number of terms. 



Hand Book of Calculations. 175 

ARITHMETICAL PROGRESSION. 

Example. 

The extremes are 3 and 29, and the number of terms 14, 
required the common difference. 

29—3=26; and 26-r- 13=2. Ans. 

Given the common difference and extremes, to find the number 
of terms. 

Rule. 

Divide the difference of the extremes by the common differ- 
ence, and to the quotient add one. 

Example. 

The first term of an arithmetical progression is 11, the last 
term 88, and the common difference 7. What is the number of 
terms? 

88 — 11 = 77; and 77^7 = 11; 11+1=12. Ans. 

Given the extremes and the number of terms, to find the sum 
of all the terms. 

Rule. 

Multiply half the sum of the extremes by the number of 
terms. 

Example. 

How many times does the hammer of a clock strike in 12 
hours. 

1+12=13 = the sum of extremes. 
Then 12x(13--2) = 78. Ans. 




If6 Hand Book of Calculations. 



WATER. 



There are some underlying natural laws and other data relating 
to water which every engineer should thoroughly understand. 
Heat, ivaier, steam, are the three properties with which he h^s 
first to deal; like the first rounds of a ladder they lead to higher 
lessons. 

The scientific head under which water is treated is Hydros- 
tatics. Hydraulics is one division of the general subject and 
means flowing water, not so applicable to steam engineering 
as the first term, which broadly means the science of fluids, of 
which water is the principal example. 

WATER AS A STANDARD. 

There are four notable temperatures for water, namely, 

32° F., or 0° C. = the freezing point under one atmosphere. 

39°. 1 or 4° = the point of maximum density. 

62° or 16°. 66 = the standard temperature. 

212° or 100° = the boiling point, under one atmosphere. 

The temperature 62° F. is the temperature of water used in 
calculating the specific gravity of bodies, with respect to the 
gravity or density of water as a basis, or as unity. 

Weight of one cubic foot of Pure Water. 

At 32° F. = 62.418 pounds. 

At39°.l =62.425 " 

At 62° (Standard temperature) =62.355 
At 212° =59.640 « 

The weight of a cubic foot of water is, it may be added, 
about 1000 ounces (exactly 998.8 ounces), at the temperature 
of maximum density. 



Ha iid Book of Calculations. if J 

WATER. 

The weight of water is usually taken in round numbers, for 
ordinary calculations, at 62.4 lbs. per cubic foot, which is the 
weight at 52°.3 F.; or it is taken at 62^- lbs. per cubic foot, 
where precision is not required, equal to L j™ lbs. 

The weight of a cylindrical foot of water at 62° F. is 48.973 
pounds. 

Weight of one cubic inch of Pure Water. 

At 32° F. = .03612 pound, or 0.5779 ounce. 

At37°.l =.036125 " " 0.5780 " 

At 62 =.03608 " " 0.5773 " or 252.595 grains. 

At 212° =.03451 " " 0.5522 " 

The weight of one cylindrical inch of pure water at 62° F. is 
• 02833 pounds, or 0.4533 ounce. 

Volume of one pound of Pure Water. 

At 32 e F. = .016021 cubic foot, or 27.684 cubic inches. 
At 39°.l = .016019 " " 27.680 

At 02 =.016037 " " 27.712 " 

At 212° =.016770 " "28.978 

The volume of one ounce of pure water at 62° F. is 1.732 
cnbic inches. 



iy8 Hand Book of Calculations. 



WATER. 



SEVERAL PRINCIPLES IMPORTANT TO KNOW. 

Water is practically non-elastic. A pressure lias been 
applied of 30,000 lbs. to the square inch and the contraction 
has been found to be less than one-twelfth. Experiment 
appears to show that for each atmosphere of pressure it is con- 
densed 47J millionth of its bulk. 

Water at rest presses equally in all directions. This is a 
most remarkable property — solids pressing only downward, or 
in the direction of gravity — the upward direction of the press- 
ure of water is equal to that pressing downwards, and the side 
pressure is also equal. 

A given pressure or bloiv impressed on any portion of a mass 
of ivater confined in a vessel is distributed equally through all 
parts of the mass; for example, a plug forced inwards on a 
square inch of the surface of water, is suddenly communicated 
to every square inch of the vessel's surface, however large, and 
to every inch of the surface of any body immersed in it. 

It is this principle which operates with such astonishing- 
effect in hydrostatic presses, of which familiar examples are 
fuund in the hydraulic pumps, by the use of which boilers are 
tested. By the mere weight of a man's body when leaning on 
the extremity ci a lever, a pressure may be produced of upwards 
of 2000 tons; it is the simplest and most easily applicable of all 
contrivances for increasing human power, and it is only limited 
by want of materials of sufficient strength to utilize it. 

The surface of water at rest is horizontal. A familiar exam- 
ple of this may be noted in the fact that the water in a battery 
of boilers also seeks a uniform level, no matter how much the 
cylinders may vary m size. 



Hand Book of Calculations. fjg 



WATER. 

T7ie pressure on any particle of water is proportioned to its 
depth below the surface, and as the side pressure is equal to the 
downward pressure, calculations on this principle are easily 
made. The pressure on a square foot at different depths are 
approximate, as in the following table. 



' Depth in 
feet 


Pressure on 
sq. foot. 


Depth in 

feet. 
♦ 

56 


Pressure on 
sq. foot. 


8 


500 lbs. 


3500 lbs. 


16 


1000 " 


64 


4000 " 


n 


1500 " 


72 


4500 " 


3*3 


2000 " 


80 


5000 " 


40 


2500 " 


88 


5500 " 


48 


3000 " 


96 


6000 <• 



1 mile, or 5/280 feet, 330,000 lbs. 
5 mf.es, 1,650,000 " 

This table is based upon an allowance of 62£ lbs. of water to 
the cubic foot, hence 8 feet X 62 \ = 500, etc. 

Wafer rises to the same level in the opposite arms of a 
recurved tube, hence water will rise in pipes as high as its 
source; this is the principle of carrying the water of an aque- 
duct through all the undulations of the ground. 

Any quantity of water, however small, may he made to lal- 
an e any quant it u, however great. This is called the Hydros- 
tatic Paradox, and is sometimes exemplified by pouring liquids 
into cask- through long tubes inserted in the bung holes. As 
- "ii as i he cask is full and the water rises in the pipe to a cer- 
tain height the cask bursts witn violence. 

JN OTE. 

The in']' regsure, dne to these peculiar natural laws, 

renders it accessary for the engineer t > make due allowances 
on the strength of pipes and v< ss< Is used for containing or con- 
voy i: 



i8o Hand Book of Calctilations. 



PUMPS. 



The action of a pump is as follows: The piston or plunder 
by moving to one end, or out of the pump cylinder, bares the 
space it occupied, cr passed through, to be filled by something. 
As there is little or no air therein a partial vacuum is formed 
unless the supply to the pump is of sufficient force to follow the 
piston or plunger of its own accord. If this is not the case, 
however, as it is where the water level from which the pump 
obtains its supply is below the pump itself, there being a par- 
tial vacuum produced, the atmospheric pressure forces the water 
into the space displaced by the plunger or piston, continuing 
its flow until the end cf stroke is reached. 

The water then ceases to flow in, and the suction valve of the 
pump closes, forbidding the water flowing back the route it 
came. The piston or plunger then begins to return into the 
space it has just vacated, and which, has become filled with 
water, and immediately meets with a resistance which would 
be insurmountable were the water not allowed to go somewhere. 

Its only egress is by raising the discharge valve by its own 
pressure, and passing out through it. This discharge valve is 
in a pipe leading to the bciler, and in going out of the cylinder 
by that route the water must overcome boiler pressure and its 
own friction along the passages. Water is inert and cannot act 
of itself; so it must derive this power to flow into the feed pipe 
and boiler from the steam acting upon the steam piston of the 
pump. The steam piston and pump piston are at the two ends 
of the same red. Therefore the steam pressure exerted upon 
the steam piston will ba exerted upon the pump piston direct. 



Hand Book of Calculations, 181 

PUMPS. 

There being no mechanical purchase in favor of the steam piston, 
it must have the greater area, otherwise one pressure would 
balance the other, and the pump would refuse to move. 

Fcr this reason, all boiler feed pumps have larger steam than 
water cylinder: generally, at least, 40 per cent, larger. 

Water will flow into a boiler when the head or height from 
which it obtains its pressure is greater than the height of a 
water column represented by the pressure within the bciler, or 
where the pressure from the water works supply exceeds the 
pressure of the steam in the boiler. 

Example. 

What horse power will be required to deliver 1,000 imperial 
gallons per minute against a pressure of 80 pounds per square 
inch, suction lift 20 feet, allowing 20 per cent, friction ? 

One-pound pressure is eepial to a head of 2.31 feet. Hence 
the total head to which the water is to be lifted will be 2.31 X 
80+20=^)1.8 feet. 

An imperial gallon weighs ten pounds and the horse power 
required I raise 1,000 imperial gallons per minute against a 
ire and suction lift equal to a head of 204.8 feet will be 
10X1,000X304.8 ^.^ H p and 
33,000 

02.00+20^ allowance for friction=74.47 II. P. 
Rtjxe to find Tin; water capacity of a steam pump per 

HOUR. 

1 . Find the contents of the pump in cubic inches, by multi- 
plying i la- area by the inches in strokes and by the fraction, it 
Is tall. 

2. Find the cubic inches of water pumped per hour, by mul- 
tiplying the contents of the pump by the strokes per minute 
and by 00 minutes. 

3. Find the number of cubic feet of water by dividing the 
cubic inch< s bv 1728. 



182 



Hand Book of Calculations. 



PUMPS. 

Example. 

How many cubic feet of water will he pumped in an liour by 
a 'pump 6 inches in diameter and 10 inch stroke, making 60 
strokes per minute, the pump being f full each stroke. Now, 
then: 



6X6= 



7854 
36 diameter squared. 



47124 
23562 



28.2744 



10 length of stroke. 



282.7440 

60 strokes. 



16964.6400 

60 minutes. 



1017^78.4000 

3 y i full. 



4)3053635. 2000^ 
12)783408.8000 



1728 <{ 12)63617.4000 



12)5301.4500 



441.7875 Ans. 441f gallons nearly. 



To find the pressure in pounds per square inch of a column 
of tvater. 

ItULE. 

Multiply the height of the column of water in feet by .434. 



Hand Book of Calculations. i8j 

PUMPS. 

Example. 
What is the pressure at the bottom of a column of water 410 
feet high ? 

440 
434 

1760 
1320 
1760 



190. tVA Ans. 191 lbs. nearly 
Xote. 
The correctness of this calculation is found by multiplying 1 
191 by 2.31. 

To find the height of a column of amter, in feet, the pressure 
being known. 

Rule. 
Multiply pressure by the pressure shown on gauge by 2.31. 

Example. 
If pressure shown on the gauge is 95 lbs. to square inch, what 
is the height of the column of water ? 
2.31 
95 



1155 
2079 

219.45 Ans. 220 ft, nearly. 

Note. 
The correctness of this result is proved by multiplying 220 
by .434. 

To find the horse power necessary to pump water to a given 
height. 

Rule. 

'Multiply the total weight of the water in pounds, by the 
height in feet, and divide the product by 33,000. 



1 8//. Hand Book of Calculations. 

PUMPS. 

Example. 
What power is required to elevate 90,800 lbs. of water 45 ft. ? 
90,800 
45 



454000 
363200 



33,000)4086000(12311 horse power. 

Note. 
This calculation allows the water to be raised in one minute. 
To raise the same amount in 60 minutes would require ^Vth the 
power. Ans. Nearly 3 horse power. 

To find quantity of ivater pumped in one minute running at 
100 feet of piston speed per minute. 

KULE. 

Square the diameter of the water cylinder in inches, and 
multiply by 4. The answer will be in gallons. 

Example. 
What quantity of water will be pumped by a 4 inch water 
cylinder with piston travelling 100 feet per minute. 
4 inch diam.=16 inches. 
4 

64 Ans. in gallons. 
Note. 
This is an approximate, not an exact quantity, as will be 
found by figuring the exact area of the piston 12.566 inches X 
100x12 inches^-231=652 6 3 4 T ; but the rule is nearer than the 
average practice of pumps, owing to leakage of air, etc. 

To find the capacity of a water cylinder of a steam pump in 
gallons. 

EULE. 

Multiply the area in inches by the length of stroke (this gives 
the capacity in cubic inches). Next divide by 231 (which is 
the cubical contents of a U. S. gallon in inches) and the pro- 
duct is the capacity in gallons. 



Hand Book of Calculations. ■ 185 

PUMPS. 

Example. 
What is the capacity of a cylinder 9 inches diameter and 10 
inch stroke. 

9 inch diameter=see table 63.617 area. 

10 

Ans. in cubic inches* 

231)636.170(2. T V\r gallons. 
462 



1741 
1617 



1247 
1155 



92 



To find the steam pressure required when the diameter of 

steam cylinder, diameter of pump cylinder, and water pressure 

are given. . 

Rule. 

Multiply the area of the pump in inches by the pressure of 
water in pounds perscpiare inch, and divide the product by the 
ana of cylinder, plus one-fourth for friction. 

Example. 
6 inches diameter of pump cylinder, 100 pounds pressure per 
square inch. 70.88 area of steam cylinder. 

" ' =39.8+i=-±9.7, nearly 50 lbs. pressure per 

« I Loo 

square inch. 

To find the diameter of cylinder required for a direct-acting 
steam pump. 

Rule. 

Multiply the area of pump-bucket or ram in inches by the 
pressure of water in pounds per square inch, and divide the 
product by the pressure of steam in pounds per square inch, 

and add one-fourth to one-half for friction. 



i86 Hand Book of Calculations. 

PUMPS. 

Example. 

6 inches diameter of pump, 100 lbs. water pressure per 
square inch, 50 lbs. steam pressure. 

6 inches diameter = 28.27 inches area, 

Kr . =56.54 inches, area of steam cylinder, 

00 

add i=56.54-fl4.13=7Gi67=9| inches diameter of steam 
■cylinder, nearly. 

To find the load on a pump: 

Kule. 

Multiply the area of pump in inches by the weight of the 
column of water in pounds per square inch. 

Example. 

Pump 3 inches in diameter;, depth of well 30 feet. 
3 inches diameter = 7.06 inches area, 

30X44 . 

1^ — = lo.2 lbs. pressure per square inch, 

7.06x13 = 91.78 lbs. total pressure on pump. 

To find the total amount of pressure that can le exerted in a 
steam pump. 

Kule. 

Multiply the area of the steam piston by the steam pressure. 

Example. 

What is the total amount of pressure in a pump cylinder 8-J 
inches diameter and 80 lbs. of steam ? 

8-J inch diam.=per table 56.745 area. 

80 lbs. to sq. in. 



4539.600 Ans. 4.539 T 6 o lbs. 



Hand Book of Calculations. 1S7 



PUMPS. 

To find the resistance of the water in the water cylinder. 

Rule. 
Multiply the area of the water piston in inches, by the press- 
ure of water in pounds per square inch. 

Example. 

What is the resistance to be overcome in a 7 inch diameter 
piston, working against a pressure of 110 lbs. 

7 in. diam.=(see table) 38.484 square inches. 

110 



384840 
384840 



4.233fWV lbs. Ans. 

To find the number of horse power required to raise a given 
quantity of water in gallons to a given height in feet. 

Kule. 

Multiply the given number of gallons of water ,to be raised 
per minute by 10 (which is the weight of one gallon) and by 
the height the water has to be raised in feet, and divide the 
product by 33,000. 

There is no account taken of lo*s by leakage or " slip," nor 
friction in any of these rules; these vary greatly according to 
the class and condition of pump, if it is working against a high 
or low lift, and the " slip " depending upon the class ( f valve 
used for the pump. 

For well designed direct-acting horizontal pumps, cue-tenth 
should be enough for " slip," for ordinary purposes, and 25 per 
cent, tor friction, but when the suction is very long and the 
height to where the water is raised is great, one-third should 
be added; but if the pump is old and badly designed, as much 
as one-half must be added to the total amount required. 

In an ordinary direct-acting steam pump one-fourth of the 
power required should be added, but if thedtlivery height is 
very great and the pipe very long one-half should be added. 



1 88 Hand Book of Calculations. 

PUMPS. 

Notes. 

The space betiveen the suction-valve and bucket, plunger or 
piston, as the case may be, should be as little as possible, con- 
sistent with ample waterway being given. 

All passages should be as straight as possible, and when bends 
are necessary, the radius should be an easy one. 

Sudden enlargements and contractions in the passages should 
be avoided, but if an alteration in size or shape of the passages 
is necessary, it should be made gradually. 

Care should be taken that the suction-pipe should be the lowest 
point in the pump. 

If the pump is required to raise hot water, there should be 
very little suction; in fact, it is best, if possible, to h aye the 
water running into the pump. 

Long suction-pipes should always be provided with a foot- 
valve just above the windbore or strainer, in the well or }3it. 

All corners should be well rounded. 

There should be as few flat surfaces as possible, and where 
there are any they should be well ribbed. 

Joints in the suction-pipes and the suction part of the pump 
must be very carefully made, and perfectly tight. 

Change of direction in the flow of water should be avoided as 
much as possible. After a current of water has received an 
impulse, it is necessary that the motion imparted should be 
continued with a uniform velocity throughout its whole course. 



Hand Book oj Calculations, 



i8g 



TABLE I. 

Quantity of Water Discharged per Mixute by Sixgle- 
Cylindeb P r.M ps, from 2 to 6-inch diameter, at 30 and 40 
strokes per minute, 0, 10 and 12 inch stroke. 





0-inch 


Stroke. 


Diameter 


Gallons per Minute. 


of Pomp. 






:» Strokes. 40 Strokes 


in. 






2 


3.0 


4.0 


2* 


4.6 


0.25 


3 


6.7 


8.93 


3* 


8.83 


12.2 


4 


11.96 


15.9 


44 


15.2 


20.3 


5 


18.75 


25.0 


54 


22.69 


30.25 


6 


27.0 


3G.0 



10-inc.b Stroke. 
Gallons per Minute. 



:0 Strokes. 10 Strokes. 

.1 



3.33 

5.21 
7.44 
9.81 
13.28 
16.88 
20.83 
25.21 
30.0 



4.44 
6.94 
9.92 
13.55 
17.66 
22,55 
iC i . i i 

33.5 

40.0 



12-inch Strok \ 
Gallons per Minute. 



:>0 Strokes. 40 Strokes. 



4.0 
6.25 
8.93 
11.77 
15.94 
20.26 
25.0 
30.25 
36.0 



5.3 
8.33 
11.9 
16.26 
21.2 
27.6 
33.33 
40.33 
48.0 



TABLE IT. 

Quantity of Water Discharged per Mixute by Double- 
Cylixder Pumps, from 2 to 6-inch diameter, at 30 and 40 
strokes per minute, 9, 10 and 12-inch stroke. 





9-inch Stroke. 


10-inch Stroke. 


13-inch Stroke. 


Diameter 


Gallons per Minute. 


Gallons per Minute. 


Gallons per Minute. 


limp. 














(0 Strokes. 


40 Strokes. 


30 Strokes. 


to Strokes. 


30 Strokes. 


K) Strokes. 


in. 
2 


0.0 


8.0 


0.00 


8.88 


8.0 


10.(5 


21 


9.38 


12.5 


10.42 


13.88 


12.5 


10 «H 


3 


13.4 


17.86 


14.88 


19.84 


17. .°fi 


23.8 


H 


17.66 


24.4 


19.02 


27.10 


23.51 


32.52 


4 


23.92 


31.8 


26.50 


35.32 


31.88 


42.4 


44 


30 4 


40.6 


33.76 


45.1 


40.52 


54.12 


5 


37.5 


50.0 


41.66 


55.54 


50.0 


60.66 


5 V 


45.38 


00.5 


50.42 


67.0 


60.5 


SO. (]0 


(J 


54.0 


72.0 


oo.o 


80.0 


72.0 


96.0 



igo 



Hand Book of Calculations. 



TABLE III. 

Pressure of Water at Different Heads in lbs. per 

Square Inch. 






c ^~ l 



10 

20 

ao 

40 

50 

60 

70 

80 

90 

100 

110 

120 

130 

140 



3.33 

Q.rS 
10.0 
13.3 
16.G 
20.0 
23.3 
26.Q 
30.0 
33.3 
36.6 
40.0 
43.3 
46.6 









1.66 
3.33 
5.0 
6.66 
8.33 
10.0 
11.6 
13.3 
15.0 
16.6 
18.3 
20.0 
21.6 
23.3 



3.041 
6.09i 
9.14s 
12.1 

15.2 
18.2 
21.3 
24.3 
27.4 
30.4 
33.5 
36.5 
39.6 
42.6 



4.33 

S.6G 
12.9 
17.3 
21.6 
25.9 
30.3 
34.6 
38.9 
43.3 
47.6 
51.9 
56.3 
60.6 



150 
160 
170 
180 
190 
200 
210 
220 
230 
240 
250 
260 
2T0 
280 



Of 


-U CO 

s s 

o -^ 


53 

SI 

eg a 


50.0 


25.0 


45.7 


53.3 


26.6 


48.7 


56.6 


28.3 


51.8 


60.0 


30.0 


54.8 


63.3 


31.6 


57.9 


66.6 


33.3 


60.9 


70.0 


35.0 


64.0 


73.3 


36.6 


67.0 


76.6 


38.3 


70.1 


80.0 


40.0 


73.1 


83.3 


41.6 


76.2 


86.6 


43.3 


79.2 


90.0 


45.0 


82.2 


93.3 


46.6 


85.3 



Ed 



64.9 

H9.3 

73.6 

77.D 

82.3 

86.5 

90.9 

95.3 

99.6 

103.9 

108.3 

112.6 

116.9 

121.3 



TABLE IY. 
Of the Diameters of Pipes, Sufficient in Size to Dis- 
charge a Kequired Quantity of Water per Minuie. 



Cubic 


Diameter in 


Cubic 


Diameter in 


Cubic 


Diameter in 


feet. 


inches. 


feet. 


inches. 


feet. 


inches. 


1 


.96 


18 


4.07 


130 


10.94 


2 


1.36 


20 


4.29 


140 


11.35 


3 


1.66 


25 


4.80 


150 


11.75 


4 


1.92 


30 


5.25 


160 


12.14 


5 


2,15 


35 


5.67 


170 


12.51 


6 


2.35 


40 


6.07 


180 


12.67 


7 


2.60 


45 


6.53 


190 


13.23 


8 


2.72 


50 


6.80 


200 


13.57 


9 


2.88 


55 


7.12 


225 


14.40 


10 


3.04 


60 


7.43 


250 


15.17 


11 


3.18 


70 


8.03 


275 


15.91 


12 


3.33 


80 


8.60 


300 


16.62 


13 


3.46 


90 


9.10 


350 


17.95 


14 


3.60 


100 


9.<;o 


400 


19.20 


15 


3.72 


i 110 


10.06 


500 


20.46 


13 


3.84 


! 120 


10.51 


<500 


23.51 



Hand Book of Calculations, iyi 



EVOLUTION OR SQUARE ROOT. 

This is one of the most important rules in the whole range 
of mathematics and well worth the careful attention of the 
student. 

Given any power of a number to find its root. To familiarize 
oneself with the extracting of the square root it is well first to 
square a number and then work backward according to the 
Examples here given, and by long and frequent practise become 
expert in the calculation. But in first working square root 
it is undoubtedly better to secure the services of a teacher. 

Example. 

Find the square root of 186624. Proof 432 

18,66,24(432 432 

16 



83 



"266" S64 

249 1296 

1728 



862 1T24 
1724 



186624 

Begin at the last figure 4, count two figures, and mark the 
second as shown in the Example; count two more, and mark 
the figure, and so on till there are no more figures; take the 
figures to the left of the last dot, 18, and find what number 
multiplied by itself will give 18: there is no number that will 
do so, for 4x4 = 16, is too small, and 5x5 = 25, is too large; 
we take the one that is too small, viz., 4, and place it in the 
quotient, and place its square 16 under the 18, subtract and 
bring down the next two figures G6. To get the divisor multi- 
ply the quotient 4 by 2 = 8, place the 8 in the divisor, and say 
H into :\ times, place the 3 after the 4 in the quotient, 

and also after the 8 in the divisor; multiply the 83 by the 3 in 
the quotient, and place the product under the 266 and subtract, 
then bring down the next two figures 24. To get the next di- 
vi« r. multiply the quotient 43 by 2=86; see how often 8 goes 
into 17, twice; place tie- 2 aftei the 43 of the quotient, and 
the 86 of tie- divisor; multiply the 862 by the 2, and 
put it under the 1 724, thei] er, 432. 



IQ2 Hand Book of Calculations. 



SQUARE 


ROOT. 






Example. 




Find the square root of 


735. 






47 


7 35(27.11 

4 

335 

329 


&c. 




Proof. 2711 
2711 

2711 


541 


600 
541 


2711 

18977 


5421 


5900 
5421 


5422 



&c. 734.9522 

We proceed as before till we get the remainder 6, and we see 
it is notSa perfect .square; we wish the root to be taken to two 
or three places of decimal; there are no more figures to bring 
down, therefore, bring down two ciphers and proceed as in the 
first Example; to the remainder attach two more ciphers and 
proceed as before; and by attaching two ciphers to the remain- 
der, you may carry it to any number of decimal places you 
please. In the above Example the answer is 27.11 &c. 

Example. 



Eind the square root of 588.0625. 

5,88.06,25(24.25 
4 


44 


188 
176 


482 


1206 
964 


484^ 


5 1 24225 
24225 



In a decimal quantity like the above, the marking off differs 
from the former Examples. Instead of counting twos from 
right to left, we begin at the decimal point and count twos 
towards the left and towards the right. The rest of the work 
is similar to the other examples. Notice, that when the .06 is 
brought down, the figure for a quotient is a decimal. 



Hand Book of Calculations. ipj 

SQUARE ROOT. 

Example. 
Find the square root of 7986.57246. 

7986.57,24 6(89.3676 &c. 
64 
169 I 1586 
1 1521 
1783 



6557 
5349 



17866 

178727" 

1787346 


120824 
107196 
1362860 
1251089 
11177100 
10724076 



453024 
Notice, the last figure is 6; always bring down two figures at 
a time, therefore bring down 60. The rest . is similar to the 
former example. 

Examples for Exercise. 
Find the square root of 589824. 



J- • 




2. 


a 


3. 


(< 


4. 


a 


5. 


a 


6. 


a 


7. 


<e 


8. 


(( 



a 


9876. 


a 


15227.56 


a 


698.532 


et 


96118416 


a 


123456 


a 


170.3025 


a 


17640.73205 



In expressing the square root it is customary to use simply 
the mark (y/) the 2 being understood. 

All routs as well as powers of one are 1, as yl = l. 

INVOLUTION 
la the raising a number (called the root) to any power. The 
powers of a Dumber are its square, cube, 4th power, 5th power, 

2x2=4 4 is the square or 2nd power of 2 

2X2X2=8 8 is tin- cube oi 3rd power of 2 

•JX2 = 16 L6 i- the 4th power of 2. 

&c. &c. 

Hence to square s Dumber multiply it by itself. 



IQ4- Hand Book of Calctdations. 

SQUARE ROOT. 
Example. 
What is the square of 27 (written 27 2 )? 

27 
27 



189 
54 



729 Answer. 
To cube a number, multiply the square of the number by 
the number again, that is, multiply the number by itself three 
times. 

Example. What is the cube of 50? (written 50 3 ) 
50 
50 

2500 the square 
50 



125000 the cube. 
The 4th, 5th, 6th, &c, power is found by multiplying the 
number by itself 4 times, 5 times, 6 times, &c, as the case may 
be. 

Example. What is the sixth power of 90? (written 90 6 ) 
90 
90 



8100 

90 


square 


729000 
90 


cube 


65610000 
90 


4th power 


5904900000 
90 


5th power 



531441000000 6th power 



Hand Book of Calculations. 195 

Ox the Signs that Represent the Roots of Numbers. 

The Sign common to all roots is \l or y and is known as 
the Radical Sign. If we require to express the square root of a 
number we simply put this sign before it, as yi6, but if the 
number is made up of two or more terms, then we express the 
square root by the same in front, but with a line as far as the 
square root extends, as \/9+7 or ^/4 (19+6). 

The cube root is expressed by the same sign, with a 3 in the 
elbow, as y8 or V" (100 — 517 

All other roots in the same manner, the number of the root 
being put instead of the 3. As fifth root y', and sixth root y ', 
&c. 

In the above examples of the square root 94-7 = 16, and the 
square root of 10 is 4. 

The 4 (19-|-6) = 4x25 = 100, and the square root of 100 is 10. 

other way of expressing that the foot is required, is by 

1 
putting a fraction after and above the quantity, as 16% which 

means the square root of 16, (19+17)*, or {4 (19+6)1* all of 

which mean the square root of the quantities to which they are 

attached. 

The cube root. 1th root, 5th root, &c., are written in the 
same way, as 729*=9; 256*=4; 3125"> = 5; &c. 

Ox Jin: Signs Repbesenjing the Power of Numbers. 

6 1 is equal to 6x6 = 36; that is, 36 is the square of 6. 

is equal to 5x5x5 = 125; that is, 125 is the cube of 5. 
V is equal to 4x4x4x4 = 256, that is, 256 is the fourth 
power of 4. 

be above we have the powers that are most frequently 

met with; but of course you may have the 5th, 6th, or any 

r: but whatever tin- power, multiply the given 

number that number of times by itself, and you will be quite 

•: for an example, what is the value of ! 7 to 

ghth power, or ! X7x7x7x7x7x7x7=5764801. 



ig6 Hand Book of Calculations. 

The power and the root are often combined, as 4*; this is 
read as the square root cf 4 cubed. So the numerator figure 
represents the power, and the denominator figure represents 
the root. In this case 4 cubed=64, and the square root of 
64=8. Answer. 

Perhaps the most common form that an engineer will meet 
with this sign is in the following: — 

8 f , which is read the cube root of 8 squared. Now 8 
squared = 64, and the cube root of 64 is 4. Answer. 

Find the value of 20*. 

20 cubed=8000; and square root of 8000=89.4 &c. 

Example. 

8*+81 

What is the value of » ? 

3* 

$*=J/8* = ^/te=4c; 81^=9; 3*= ^¥~=^/2V=5.2 nearly. 
4+9 13 t A 

Hence, ~X~2 == ^¥ = ^'^ or * Answer. 

( ) are called brackets, and mean that all the quantities 
within them are to be put together first; thus, 7 (8 — 6+4x3) 
means that 6 must be subtracted from 8=2, and 4 times 3=12 
added to this 2=14; and then this 14 is to be multiplied by 

7=98. 

CIRCULAR INCHES. 

A circular inch is a circle whose diameter is one inch; instead 
of finding diameters in square inches it is frequently conveni- 
ent to use the circular inch as per the following 

Rule. 

To find the circular inches in a circle: square the diameter 
of the circle in inches. 

Examples. 

1. How many circular inches are there in a safety valve 
whose diameter is 4J- inches ? 

4.5 2 = 4.5 X 4.5 = 20i Answer. 



HcuiiJ Book of Calculations. 197 

CIRCULAR INCHES. 

'2. How many circular inches in a compound engine whose 
diameters are 31' and 60' ? 

Answer. 31 8 and 60 8 = 4561 circular inches. 

3. The diameter of a piston is 24 inches, how many circular 
inches will that give ? Now 

24 a = 24x24 = Ans. 576 circular inches. 

4. If two pistons of a compound engine are 26" and 50", 
what ratio will their areas hear to each other ? 

Instead of finding the areas in square inches find them in 
circular inches. 

26 2 — 676, and 50 2 = 2500. 
Hence as 676 : 2500:: 1 : Answer, 1 to 3.7 nearly. 



COMPOSITION OF AIR, ETC. 

Air is composed of nitrogen and oxygen mixed mechanically 

and not chemically, (this is unlike water, the parts of which are 

combined chemically.) Out of every 100 volumes of air 79 

parts are composed <f nitrogen to 21 of oxygen, or by iveiglit, 

1 to 23 cf oxygen. 

150 cubic feet of air are necessary to burn 1 lb. of coal, but 
in practice double that quantity should be supplied to the 
furnace. 

12 lbs. of air are required to burn 1 lb. of carbon. 

36 lbs. of air are ne ■< ssary to burn 1 lb. of hydrogen. 

Bydrogen while burning is A \ times hotter than oxygen. 

Hydrogen gives ont more heat from the coal — part for part 
— but as tie ttuch more of the carbon we get the great- 

est amount of h ai from it. 

;d will not bum without an admixture of air. 



ig8 Hand Book of Calculations. 



STEAM. 



Steam, properly socalled, is perfectly transparent and color- 
less. The engineer and general reader have thus alike to bear 
in mind, that in dealing with steam (proper) they have to do 
with a gaseous body which eludes the sight as completely as the 
purest atmospheric atmosphere. 

In popular language, the visible mist forming when a vapor 
is discharged into the air, as a little way from the spout of a 
boiling kettle, or in a dense cloud above an engine " blowing 
off" steam, is also called steam. This visible mist is however 
really of the nature of a cloud formed from the water, con- 
densed from the vapor and enclosing minute particles of air, 
constituting an opaque and visible mass. 

Perfect steam is, moreover, in no way moist, but is dry ; the 
moisture sometimes showing upon a solid surface it touches, or 
that has been plunged into it being due to condensation. 

The distinguishing properties of steam are, 1, Its fluidity; 2, 
its mobility; 3, its elasticity, and 4, its equality of pressure in 
every direction; that is, steam has a flow like water, it has a 
circulation within its own body, it is capable of compression 
and expansion,- and when it is confined it presses equally upon 
all parts of the restraining vessel. 

But perhaps the rapidity with which, at a given condition 
and temperature, it can be condensed and again formed makes 
its most useful property; while the cheapness and abundant 
supply of water from which steam is formed must not be over- 
looked. 

Each atom of steam is composed of two gases which have 
neither taste nor color. An atom of steam is made up of the 
same materials and in the same proporti :n as an atom of water, 
i, e., in volume one part of oxygen to two of hydrogen, but in 
weight 89 of oxygen to 11 of hydrogen. In the safe opera- 



Hand Book of Calculations. igg 

STK. 

tion of steam production theso proportions do not and cannot 
change but by contact with certain substances, such as li 
baryta, strontium, soda and potash; and at the temperatun of 
red-hot iron by contact with carbon, chlorine, phosphorus, 
iodine, zinc, tin, manganese and iron. The oxygen of the 
water forms a combination with these metals while the hydro- 
gen - e. 

This pr< ss of separation of the two gas s is called disinte- 
ud is a chemical, not a mechanical change. 

The difference in volume between water and steam at atmos- 
pheric pressure is as 1669 to 1; that is, a given quantity of 
water expanded into steam will occupy 1GG9 times the space it 
did before. 

This is nearly one cubic foot and one cubic foot of steam at 
atmospheric pressure weighs .038 lbs. 

SATURATED STEAM. 

But a limited and gradual supply of heat can under any 
circ \ made to enter water; its rate of absorption 

beii Le process prolonged; and this is 

fortunate, in view of the risk, otherwise, of continual explo- 
The tumultuous vaporizing of water is what we call 
boiling and there is a never-ceasing balancing between the heat 
and pressure within the steam \ issel for the following reason: 

■ in a boiler are, so to say, at an equipoise; 
increase of I: increase the quantity of water vaporized, 

and so, iu a confined place, the density of the vapor; or on the 
other 36 of pressure will compel a portion of the 

vapor already formed to resume the liquid state, hence a 
perpetual • dancing of the two conditions. 

Where the steam is produced over, or in communication with 

the water of a boiler, pari of its density is produced by the 

less finely divided wati r. or mist, held in 

susp trough it, hence the term "saturated sham." 

of Loosely beld vapor varies, as has been 

explai cording to temperature and pressure. 



200 



Hand Book of Calculations. 



SATURATED STEAM. 

By means of recorded observations of experiments on steam, 
and finding the mean of the most trustworthy results and cal- 
culations, some of which are intimated rather than detailed, 
very full tables of the properties of saturated and of superheated 
steam have been prepared. Of such a table for saturated steam, 
a brief abstract only can here be introduced. 

The following table gives the properties of steam at different 
pressures — from 1 lb. to 300 lbs. "total pressure/' i. e., above 
vacuum. 

The gauge pressure is about 15 pounds less than the total 
pressure, so that in using this table, 15 must be added to the 
pressure as given by the steam gauge. 

Table of Peoperties of Saturated Steam. 



Total 












Relative 


pressure 
■ per 
square 
iueh. 


Tempera- 
ture in 
degrees. 


Total heat 
in degrees 
above 32°. 


Latent heat 
in degrees. 


Density or 
weight per 
cubic foot. 


Volume of 
1 lb. of 
steam. 


volume or 
Xo. of cubic 
feet of steam 

from 1 of 
water. 














Ratio of 


Lbs. 


Fahr: 


Fahr. 


Fahr. 


Lb. 


Cubic feet. 


volume. 


1 


102.1 


1,112.5 


1,042,9 


.0030 


330.36 


20,582 


10 


193.3 


1,140.3 


978.4 


.0264 


37.84 


2,358 


14 


209.6 


1,145.3 


966.8 


.0362 


27.61 


1,720 


14.7 


212.0 


1,146.1 


965.2 


.0380 


26.36 


1,642 


15 


213.1 


1,146.4 


964.3 


.0387 


25.85 


1,610 


18 


222.4 


1,149.2 


957.7 


.0459 


21.78 


1,357 


21 


230.6 


1,151.7 


951.3 


.0531 


18.84 


1,174 


24 


237.8 


1,153.9 


946.9 


.0601 


16.64 


1,036 


30 


250.4 


1,157.8 


937.9 


.0743 


13.46 


838 


36 


260.9 


1,161.0 


930.5 


.0881 


11.34 


707 


45 


274.4 


1,165.1 


920.9 


.1089 


9.18 


•572 


60 


• 292.7 


1,170.7 


908.0 


.1425 


7.01 


437 


75 


307.5 


1,175.2 


897.5 


.1759 


5.68 


353 


90 


320.2 


1,179.1 


888.5 


.2089 


4.79 


298 


105 


331.3 


1,182.4 


880.7 


.2414 


4.14 


257 


120 


341.1 


1,185.4 


873.7 


.2738 


3.65 


227 


135 


350.1 


1/188.2 


867.4 


.3060 


3.27 


203 


150 


358.3 


1,190.7 


861.5 


.3377 


2.96 


184 


180 


372,9 


1,195.1 


851.3 


.4009 


2.49 


155 


210 


386.0 


1,199.1 


841.9 


.4634 


2.16 


135 


240 


397.5 


1,202.6 


833.8 


.5248 


1.90 


119 


270 


407.9 


1,205.8 


826.4 


.5868 


1.70 


106 


300 


417.5 


1,208.8 


819.6 


.6486 


1.54 


96 



Hand Book of Calculations. 201 



SUPERHEATED STEAM. 

If the application of heat be continued after the steam has 
been removed from the contact with the water in the vessel, or 
the water has all been evaporated into steam, the state of satu- 
ration is left behind. The steam so separated and heated loses 
the moisture which may accompany it in the saturated state, 
and at a few degrees of added temperature acquires in full the 
character of a perfect gas; it is then said to be surcharged with 
heat. 

Let steam in this condition be replaced in contact with the 
water in the boiler, or in any way be brought into free com- 
munication with it, the water having yet the original tempera- 
ture and such steam would immediately evaporate and absorb 
a further portion of the water, transferring to this its excess of 
heat, and would become saturated, its temperature falling to 
that of the water. 

Note. 

Many conditions are described and problems stated where the 
term '-saturated steam " is used; so let it be twice remembered 
that this expression denotes the regular condition of steam 
formed over water, and that such steam stands both at the con- 
densing point and at the generating point, that is, it is con- 
densed if the temperature falls and more water is evaporated if 
tin- temperature rises. 

FORMATION OF STEAM UNDER PRESSURE. 

At the sea level air has been found to weigh 14^ lbs. per 
square inch, that is, with a surface like the piston of a cylinder 
with no air (1. e. a perfect vacuum) upon the other side, the 
poshing force is 14.7 lbs. for each square inch which the piston 
measures. 

In making steam, this pressure of 14.7 lbs. to the square 
inch is overcome when water has beeu heated to 212° F. 

At this temperature and with a sufficiently hot fire the mass 
of water Larger or Bmaller changes to steam and passes into the 
atmosphere. 



202 Hand Book of Calculations. 



FORMATION OF STEAM UNDER PRESSURE. 

If, however, the pressure on the surface of the water is 
increased to the weight of, say, 1 \ atmospheres, then steam only 
begins to form at 234°. 

At 2 atmospheres pressure, steam forms at 250°; at 2-J, 264°; 
at 3, 274°; at 4, 292°; at 5, 306°; at 8, 340°; at 10, 35?%* at 
15, 389°; at 20, 415°, or about 294 lbs. per square inch. It 
need scarcely be said that this increased pressure on the water 
is caused by the confinement of the steam after its formation. 

It will also be observed that the temperature rises more 
slowly than the pressure. For example,- the pressures being 
advanced 5 lbs. per square inch, thus: 

lib. 6 lbs. 11 lbs. 16 lbs. 21 lbs. 26 lbs. 31 lbs. 36 lbs. 
the temperatures in Fahrenheit degrees are 
102°ir 170°^ 197°8 216°3 230°6 242°3 252°2 260°9 

LATENT HEAT OF STEAM. 

It is impossible for steam to exist without heat. Heat imparts 
to water that repellant force we call expansion; in effect the 
heating of the water causes each particle to repel and antago- 
nize and drive to the greatest possible distance every other par- 
ticle of the mass. 

In heating water a certain proportion of the heat which has 
been absorbed, is not shown by the thermometer or by touch, 
and there are two sorts or conditions of heat in the process of 
steam production operating upon water, 1 Sensible heat, 
2 Latent or insensible heat; hence the constituent, or total heat 
of steam consists of its latent heat in addition to its sensible 
heat. In generating water into steam there is absorbed about 
five and one-half times as much heat as is required under 
atmospheric pressure, to raise the temperature of the water 
from freezing point, 32° Fah., to boiling point, 212° Fah,, an 
amount of heat which if the water were a fixed solid would, it 
is said, render it red hot by daylight. Tested by a thermome- 
ter the steam will show only 212°, but by exj^eriment 1000° 
nearly, have been added, which is stored up in some hidden 
unaccountable way and is called the latent heat of steam. 



Hand Book of Calculations. 203 

LATENT HEAT OF STEAM. 

In calculations the expression "the total heat" represents 
units of heat when the weight of the steam is one pound. 

\ound of steam is the same as a pound of water, i. e., a 
po'und of wal srted into steam still weighs 1 pound. 

To trace the appropriation of all the heat that goes to the 
formation of a pound of steam, in the sensible and the latent 
E heat units, as well as of foot-pounds, take for 
example 1 ne pound of water at 22° Fahrenheit, and convert it 
into £ l at 212° Fahrenheit. The first instalment 

of he: t is tli s< osihle heat, and it is required for elevating the 
teie; the water to 212 , through ISO , which appro- 

priates 180.9 units of heat, equivalent to 180.9x77'-?, or 
pounds. (One heat unit = 772 foot-pom:) els.) 

ndly, latent heat is applied in overcoming the molecular 
attra nd separating the particles; that is to say, in the 

formation of steam, which appropriates 89*2.9 units of heat 
>ot-pounds. 

Thirdly, latent heat is applied in repelling the incumbent 

pressi r . vhi ther 1 f the atmosphere or of the surrounding 

: , in raising a load of 14.7 lbs. per square 

incl nil a square foot, through a cubic space of 

ibic feet, being the volume of one pound of saturated 

111. The work thus done is equal to 211G.4x2G.3G, or 

55,788 :' ot-pounds, or its equivalent, 72.3 units of heat. 

the above appropriations of the heal was 
found by subtracting the sum of the firsi and third, which are 
. arrived ai by din irvation, from the total leal. 

r I'h.- firsi appropriation of heal is thus eeeu to bo the sensible 
heat, ;ii -"nd and third together constitute the latent 

heat. . it may be added, is Bimply an expression of 

mechanical 1. b r neces ary to disengage 26.36 cubic feet 
of-' ace against an atmospheric press- 

ore 01' 2 1 LG. \ lbs. per Bquare loot. 



204 



Hand Book of Calculations. 



The appropriation of the heat expended in the generation of 
one ponnd of saturated steam at 212° F., from water supplied 
at 32° F., may be exhibited thus: — 

To Generate one pound oe Steam at 212° F. 





Units of heat. 


Mechanical equivalent 
in foot-pounds. 


The sensible heat: — 






1 . To raise the tempera- 






ture of the water 






from 32° to 212° F., 


180.9 


139,655 


The latent heat : — 






2. In the formation of 






steam 


892.935 


689,346 


3. In resisting the in- 






cumbent atmospher- 






ic pressure of 14.7 






lbs. per square inch, 






or 2116.4 lbs. per 






square f oot 


72.265 


55,788 


Total or constituent heat. . 


965.2 


<y\*\ 131 




1146.1 


881,789 



EULE EOR FINDING THE TOTAL HEAT IN STEAM. 

Multiply the temperature or sensible heat of the steam by .3 
( T 3 oths) and add it to 1115°. 

Example. 
Find the total and latent heat in steam that is 60 lbs. by the 
gauge. 

60 lbs. by the gauge is equal to 75 lbs. gross (the 14.7 atmos- 
phere being added, as, in gross 15 lbs.) 

And 75 lbs. gross has 307° temperature (see Table), hence, 
307° x .3 = 92.1+1115° = 1207.1 total heat. 
1207.1= total heat. 
307 = sensible heat. 



900.1 latent heat. 
Then, if we know the temperature of the feed water, arid 
subtract this temperature from the total heat of the steam, 
the remainder will be the units of heat to each lb. of 
water turned into steam; to illustrate this see the following 



Hand Book of Calculations. 205 

Examples. 

What are the total units of heat in steam of 212° 

212°X.3=63.6+1115°==1178.6 total heat. 
What is the latent heat in this case? 
1178.6=total heat. 
212 =sensible heat. 



966. 6=latent heat. 
Example. 

If the steam in the boiler be 270° and the feed water be at 
110° how many units of heat will it be necessary to add to this 
water to turn a lb. of it into steam ? 

■! 70X.3 — 81+1115 — 1196, less feed water 110 = 1086 Ans. 

Note. 

The small variation between the results in the examples and 
the figures in the Table is caused by greater detail of calcula- 
tion in one more than the other. In the examples the air 
pressure is extended at 15 lbs. per square inch and in the 
Tables at 14.7. 

Let it be remembered that a Thermal unit (expressed by T. 
U.) is the raising of 1 lb. of water 1 degree, and that the 
mechanical force existing in each unit is 772 lbs. 

OUTFLOW OF STEAM THROUGH AN ORIFICE. 

The velocity of steam escaping from under pressure is known 
to be very great though few are aware that even under a mod- 
erate pressure of Bay twenty or thirty pounds to the square 
inch, it is equal to that of a projectile fired from a cannon. 

A notable example of the high velocity of escaping steam 
Lb thai of a steam whistle, in which a jet of steam little thicker 
than ordinary writing paper, produces a sound that can be 
heard further than the loudest thunder; a railroad whistle 
has often been heard eighteen to twenty miles, while thunder 
is seldom heard over ten or twelve inilea. Every engineei 
knows how little his safety valve lifts, while the whole current 
of .-team required to run his engine escapes therefrom. 



206 



Hand Book of Calculations. 



Steam acts like a liquid in flowing through openings and 
tubes; and the velocity of flow is regulated by the same funda^ 
mental laws that govern liquids. 

But, while the height through which the water falls can be 
ascertained by direct measurement, for steam it is necessary 
to make calculations. The following two tables embrace the 
results both of the figures and practical tests. 

Velocity of Efflux of Steam into the Atmosphere. 



Absolute 
initial press- 
ure per 
square 
inch- 


Outside 

pressure 

per square 

inch. 


Ratio of 
expansion 
of nozzle. 


Velocity of 
efflux, as at con- 
stant density. 


Actual A r eloci- 
ty of efflux, 
expanded. 


Weight of 
steam dis- 
charged per 
minute per 
square inch. 


lbs. 


lbs. 


ratio. 


feet per second. 


feet per second. 


pounds. 


25.37 


14.7 


1.624 


863 


1401 


22.81 


30 


14.7 


1.624 


867 


1408 


26.84 


40 


14.7 


1.624 


874 


1419 


35.18 


45 


14.7 


1.624 


877 


1424 


39.78 


50 


14.7 


1.624 


880 


1429 


44.06 


60 


14.7 


1.624 


885 


1437 


52,59 


70 


14.7 


1.624 


889 


1444 


61.07 


75 


14.7 


1.624 


891 


1447 


65.30 


90 


14.7 


1.624 


895 


1454 


77.94 


100 


14.7 


1.624 


898 


1459 


86.34 



Outflow of Steam: — From a given" Absolute Initial 

Pressure into Various Lower Pressures. 

Initial Pressure in Boiler^ 75 lbs. per square inch. 



Absolute 
pressure in 
boiler per 

square 

inch. 


Outside press- 
ure per square 
inch. 


Rate of 
expansion 
in nozzle. 


Velocity of 
efflux as con- 
stant density. 


Actual veloc- 
ity of efflux 
expanded. 


Weight dis- 
charged per 
square inch 
of orifice per 
minute. 


lbs. 


lbs. 


ratio. 


feet per second 


feet per second 


pounds. 


75 


74 


1.012 


227.5 


230 


16.68 


75 


72 


1.037 


386.7 


401 


28.35 


75 


70 


1.063 


490 


521 


35.93 


75 


65 


1.136 


660 


749 


48.38 


75 


61.62 


1.198 


736 


876 


53.97 


75 


60 


1.219 


765 


933 


56.12 


75 


50 


1.434 


873 


1252 


64. 


75 


45 


1.575 


890 


1401 


65.24 


75 


j 43.46 ) 

| (53 p. ct.) f 


1.624 


890.6 


1446.5 


65.3 


75 


15 


1.624 


890.6 


1446.5 


65.3 


75 





1.624 


890.6 


1446.5 


65.3 



Hand Book of Calculations: 207 

VELOCITY OF STEAM. 

Note. 
Practically the results do n I agree exactly with the Tables. 
There is some w; »wer from friction at the point of dis- 

charge. Ii' harge pi] e is •• Lort its length being no 

no more than its diameter and properly enlarged inside, there 
be but little loss of power, whereas if the steam escapes 
through a pi pc of considerable length, the steam will expand 
able in passing its length, and while thus expand- 
ing back pressure on that back of it; thus retarding 
fcho velocity of that just entering the pipe and rendering the 
of steam correspondingly less. 

VALUE OF EXHAUST STEAM. 
Owing principally to its latent heat exhaust steam is of prac- 
tically the same value as an equal quantity of direct steam of 
• fi r heating in the winter season, and- with proper 
arra ts, for many of the numberless operations carried 

on in textile and other manufactories. For ail these purposes 
ipes must necessarily be somewhat larger than they need 
. here direct steam of high pressure is used. In many cases 
wh re f ailur suited from an attempt to use exhaust for 

the above j be result has been due to the use of a too 

3ystem of piping, and in other cases, to a wrongly 

in. 

e demand for steam for which the exhaust may be 

; he supply of exhaust is variable, automatic 

!• making up the requisite quantity direct from 

construct* d, so as to always insure the 

sonomy. When more heat than the 

exhaust can furnish is wanted, the proper amount needed to 

•d is drawn from the boilers; when the exhaust 

fun. ire than ie needed, it can be utilized to heat feed- 

wat- r. or it can be he mi a storage tank for hot water. 

K\I*A\ OF STEAM. 

With open porta I he e :pansiv< from the 

water in the boiler ae I point or fulcrum, but after it is 

i off " ii ads of the cylindi 



208 



Hand Book of Calculations. 



EXPANSION OF STEAM. 

When steam first enters the cylinder, the space it exists in 
becomes enlarged by that through which the piston moves, and 
again the supply from the boiler to the cylinder is i{ cut oil " at 
an early period in " the stroke/' not merely for avoiding waste, 
or to assure smoothness of action, but as a positive means for 
increasing the economical performance of the steam; hence the 
whole process carried on within the cylinder is one of expan- 
sion. 

The beneficial results accruing from the use of steam, expan- 
sively, and the methods of calculations relating to it will be 
considered hereafter, under the heading The Indicatoe. 

Table, 
nljmbee oe thebmal units ik one pound of watee. 



\ Tempera- 


Number of 


Tempera- 


Number of 


Tempera- 


Number of 


ture. 


thermal units. 


ture. 


thermal units. 


ture. 


thermal units. 


35° 


35. 


150° 


150.305 


265° 


266.774 


40 


40.001 


155 


155.339 


270 


271.878 


45 


45.002 


160 


160.374 


275 


276.985 


50 


50.003 


165 


165.413 


280 


282,095 


55 


55.006 


170 


170.453 


285 


287.210 


60 


60.009 


175 


175.497 


290 


292.329 


65 


65.014 


180 


180.542 


295 


297.452 


70 


70.020 


185 


185.591 


300 


302,580 


75 


75.027 


190 


190.643 


305 


307.712 


80 


80.036 


195 


195.697 


310 


312.848 


85 


85.045 


200 


200.753 


315 


317.988 


90 


90.055 


205 


205.813 


320 


323.134 


95 


95.067 


210 


210.874 • 


! 325 


328.284 


100 


100.080 


215 


215.939 


330 


333.438 


105 


105.095 


220 


2-a. 


335 


338.596 


110 


110.110 


225 


226.078 


340 


343.759 


115 


115.129 


230 


231.153 


345 


348.927 


120 


120.149 


235 


236.232 


350 


354.101 


125 


125.169 


240 


241.313 


355 


359.280 


130 


130.192 


245 


246.398 


360 


364.464 


135 


135.217 


250 


251.487 


365 


369.653 


140 


140.245 


'255 


256.579 


370 


374.846 


145 


145.275 


260 


261.674 


375 


380.944 



Hand Book of Calculations. 209 



INJECTORS. 



This appliance, which was invented by Giffard, is in many 
respects the most peculiar and interesting apparatus connected 
with the steam engine. It is an instrument which converts 
the energy of the steam into mechanical work without the aid 
of any moving mechanism whatever. 

Before describing it, it is necessary to notice the difference 
between the velocity of steam escaping from a boiler, and water 
issuing from the same vessel under the same pressure of steam. 
It is impossible to give, in reasonable limits, an account of the 
theory and the rules by which it -is determined, but it will 
suthce to say, that for the pressure usual to land boilers, the 
velocity of the steam is from 16 to 18 times, or even more, than 
that of water. 

Suppose now that some of the steam were discharged from a 
boiler through a pipe at this high velocity, and that while in 
the act of discharge, it were condensed suddenly by passing 
through an intensely cold medium; the resulting water con- 
densed from the steam would travel forward with the same 
velocity which it had already acquired when in the state of 
steam; and if the various particles of water could by any means 
be gathered together into a continuous stream they would be 
more than able to overcome and force back into the boiler any 
opposing stream of water of the same size directed against them 
from the water room of the boiler. 

Now the velocity of the condensed steam is so great that it 

jfl not only energy enough to re-enter the boiler in the 

face of an opposing stream of water of its own size, but it can 

also impart energy to a much larger mass of water, so that this 

larger mass can also enter the boiler. 

The injector L8 simply an instrument for allowing steam to 
rush from a boiler, and lo Mick up and mix for itself a stream 
of cold water, by which it is condensed, and to which it imparts 
so much of its own velocity that the combined mass of cold 
water and condensed steam enters into and feeds the boiler. 



2IO 



Hand Book of Calculations. 



Fig. 108 shows an elementary form of such an injector. A 
is the section of a boiler, B a pipe leading from the steam space 
and terminating in a nozzle,, G is the cold water pipe leading 
from the tank, and terminating in a hollow cone surrounding 
the steam nozzle. When the steam is turned on, and escapes 
from the lower edge E of the hollow cone, it creates a partial 
vacuum in the cone and in the pipe G. The water then rushes 
up the pipe and into the cone surrounding the nozzle, when it 
meets with the escaping steam, which it condenses. 

The particles of con- 
densed steam, mingling 
with the water surround- 
ing them, communicate 
their motion to the latter, 
and the combined mass is 
delivered with a high ve- 
locity into the feed pipe 
F and through the valve 
at G into the boiler. Such 
an injector, if properly 
w proportioned, would work 

°* * well for a fixed pressure 

of steam in the boiler, and for a fixed temperature of the feed 
water. In practice however these quantities vary, and injectors 
must be made to suit all such contingencies. For instance, 
when the pressure of the steam increases, the area in the 
opening of the steam nozzle must be increased and when the 
pressure decreases it must be made smaller. 

There are very many forms of injectors. Fig. 109 illustrates 
one which is in quite common use. The steam and 
water supply pipes, nozzle and cone are rendered sufficiently- 
clear by the drawing. The steam supply is varied by altering 
the position of the conical spindle a, which can be screwed 
towards or away from the mouth of the nozzle. 

The water chamber CG is so arranged that it completely 
surrounds the steam nozzle. The supply of the water is varied 
by contracting or expanding tbe conical aperture below the 
mouth of the steam nozzle. This is accomplished by moving 
the conical sliding tube E backwards or forwards by means of 
the handwheel D and the rack and pinion. 




Hand Book of Calculations. 



211 



INJECTORS. 

If the supply of steam is not 
properly adjusted to the water, 
some of the latter will eseape at the 
aperture made in the sliding tube 
E into the overflow pipe. For 
instance, if the supply of steam be 
too small, the current will not have 
sufficient energy to enter the boiler, 
part of it will choke up the sliding 
tube and escape by the aperture. 
When this occurs it is only neces- 
sary to turn on more steam or shut 
off some of the water. 

The efficiency of the injector is 
measured by the temperature of the 
current of feed water as it enters 
the boiler, compared with its tem- 
perature before it entered the in- 
jector. The less the rise in tem- 
perature, the more the energy of 
the Bteam is utilized. 

Injectors are also used for other 
purposes besides feeding boilers. 
They are used to pump out cisterns 
and drain basins and have even 
served to pump out mines. In the 
latter case 80 gallons per minute Fig. 109. 

have been raised ^40 feet, which is 

probably the greatest amount of work which has been done by 
an injedor. 

Note. 
Fn feeding a steam boiler with water two things are necessary: 
there is a certain amount of mechanical work to be done 
in forcing the water into the boiler. Second, the water is to 
be heated; both require a certain number of heat units and in 
the injector all the steam used for forcing the water is doubtless 
economically used, but that used, in heating the water may be 
considered a waste, as compared to the use of the exhaust steam 
in a feed water heater. 




212 Hand Book of Calculations, 



GRAVITY. 



We can not say what gravity is, but what it does, — namely, 
that it is something which gives to every particle of matter a 
tendency toward every other particle. This influence is con- 
veyed from one body to another without any perceptible inter- 
val of time. If the action of gravitation is not instantaneous, 
it moves more than fifty millions of times faster than light. 

Gravity extends to all known bodies in the universe, from 
the smallest to the greatest; by ifc all bodies are drawn toward 
the center of the earth, not because there is any peculiar prop- 
erty or power in the center, but because, the earth being a 
sphere, the aggregate effect of the attractions exerted by all its 
parts upon any body exterior to it, is such as to direct the body 
toward the center. 

This property discovers itself, not only in the motion of fall- 
ing bodies, but in the pressure exerted by one portion of matter 
upon another which sustains it; and bodies descending freely 
under its influence, whatever be their figure, dimensions or 
texture, are all equally accelerated in right lines perpendicular 
to the plane of the horizon. The apparent inequality of the 
action of gravity upon different species of matter near the sur- 
face of the earth arises entirely from the resistance which they 
meet with in their passage through the air. When this resis- 
tance is removed, (as in the exhausted receiver of an air-pump,) 
no such inequality is perceived; bodies of all kinds there 
descend with equal velocities; and a coin, a feather, and the 
smallest particle of matter, if let fall together, are observed to 
reach the bottom of the receiver exactly at the same instant. 

The weight of a body is the force it exerts in consequence of 
its gravity, and is measured by its mechanical effects, such as 
bending a spring. We weigh a body by ascertaining the force 
required to hold it back, or to keep it from descending. Hence, 
weights are nothing more than measures of the force of gravity 
in different bodies. 



Hand Book of Calculations. 



213 



GRAVITY. 

It lias been ascertained, by experiment, that a body falling 
freely from rest, will descend I61V feet in the first second of 
time, and will then have acquired a velocity, which, being con- 
tinued uniformly, will carry it through %t\ feet in the next 
second. Therefore, if the first series of numbers be expressed 
in seconds, 1", 2", 3", &c, the velocities in feet will be 32£, 
64J, 9'U. &c. ; the spaces passed through as I61V, 64£, 144£, &c., 
and the spaces for each second, 16r?, 48J, 80xV, &c. 

Table 
Shouting the Relation of Time, Space, and Velocity, 



Time in 
Beconds of 

tin- body's 
fall. 


Velocity ac- 
quired at 

the end <>t' 
that time. 


Squares. 


Space fallen 

through • 

in that 

time. 


Space. 


Whole space 
fallen through 
in the last sec- 
ond of the fall. 


1 


32.16 


1 


16.08 


1 


16.08 


2 


64.33 


4 


64.33 


3 


48.25 


3 


96.5' 


9 


144.75 


5 


80.41 


4 


128.66 


16 


257.33 


7 


112.58 


5 


ICO. 83 


25 


402.08 


9 


144.75 


6 


193. 


36 


579. 


11 


176.91 


; 


223.17 


49 


788.08 


13 


209.08 


8 


257.33 


64 


1029.33 


15 


241.25 


9 


289.5 


81 


1302.75 


17 


273.42 


10 


321. 6G 


100 


1946.08 


19 


305.58 



Rule I. 

To fin 'I the velocity a falling body will acquire in any given 
time* 

Multiply tli'' time, in seconds, by 32?, and it will give the 
velocity acquired in feet, per second. 



Example. 

Required the velocity in 7 seconds. 

7 — 2251 feet. Ans. 



2T4 Hand Book of Calculations, 

GRAVITY. 
EULE II. 

To find the Velocity a Body will acquire by falling from any 
given height. 

Multiply the space, in feet, by 64-J-, the square root of the 
product will be the velocity acquired, in feet, per second. 

Example. 
Eequired the velocity which a ball has acquired in descend- 
ing through 201 feet. 

64^x201 = 12931; ^12931 = 113.7 feet. Ans. 

Rule III. 
To find the Space through which a Body ivillfall i?i any given 
time. 

Multiply the square of the time, in seconds, by 16tV, and it 
will give the space in feet. 

Example. 
Required the space fallen through in seven seconds. 
16tV X T = 788tV feet. Ans. 

RtJLE IV. 

To find the Time which a Body will ~be in falling through a 
given space. 

Divide the square root of the space fallen through by 4, and 
the quotient will be the time in which it was falling. 

Example. 
Required the time a body will be in falling through 402.08 
feet of space. 

V402, 08 = 20.049, and 20.049-=-4 = 5.012. Ans. 
Rule V. 
The Velocity being given, to find the Space fallen through. 
Divide the velocity by 8, and the square of the quotient will 
be the distance fallen through to acquire that velocity. 

Example. 
If the velocity of a cannon ball be G60 feet per second, from 
what height must a body fall to acquire the same velocity? 
660-^-8 = 82. 5 2 = 6806.25 feet. Ans. 



Hand Book of Calculations. 215 

SPECIFIC GRAVITY. 

Rule VI. 
To find the Time, the Velocity per second being given. 

Divide the given velocity by 8, and one-fourth part of the 
quotient will be the answer. 

Example. 
How long must a bullet be falling to acquire a velocity of 
480 feet? 

480-T-8 = 60-i-4 = 15 seconds. Ans. 

SPECIFIC GRAVITIES OF BODIES. 

Every substance in nature has, under the same circum- 
stances, a weight specific or peculiar to itself. 

The Specific Gravity of a body is its weight compared with 
the weight of another body taken as a standard. 

Water is the standard for all solids a, id liquids, and common 
air is the standard for gases. 

The heaviest of all known substances is platinum, whose 
specific gravity, in its state of greatest condensation, is 22, 
water 1; and the lightest of all weighable bodies is hydrogen 
gas, whose specific gravity is iV.;<>, common air being 1, but air 
is 818 times lighter than water. Hence by calculation it will 
be found that platinum is about 247,000 times as heavy as 
hydrogen and a wide range is allowed, to the various bodies 
which lie between these extremes. 

cific gravity of a liquid is usually taken by means of a 
specific gravity hoftle, graduated so as to contain exactly 1000 
grains of pure water. If this be filled with spirits of wine and 
weighed in a balance, (together with a counterpoise for the 
weight of the bottle, which of course is constant,) it will weigh 
considerably less than LOO0 grains; in fact, the bottle will con- 
tain only aboui 911 grains of proof spirit; therefore, taking the 
specific gravity of water as unity, 1 or 1.000, the specific gravi- 
ty of spirits of wine La 0.017. If, on the other hand, the bottle 
be tilled with sulphuric acid, it will weigh about 1850 grains; 
hence, the specific gravity of sulphuric acid is said to be 1.850. 



216 Hand Book of Calculations. 

SPECIFIC GRAVITY. 

In taking the specific gravity .of solids., advantage is taken of 
the important fact that when a solid is wholly immersed in 
water, it displaces a bulk of that liquid exactly equal to its 
own; and the solid appears to lose its weight; that is, it is sup- 
ported by the surrounding water with a force exactly equal to 
the weight of the water displaced; hence, the difference of its 
weight in water from that of its weight in air must be the 
weight of an equal bulk of water. 
Rule foe finding the Specific Gravity of a Solid Body. 

Weigh the solid in air and then in pure water. 

The difference is the weight of water displaced and whose 
specific gravity is 1.000. 

Then, as the difference of weight is to 1.000, so is the weight 
in air to the specific gravity; or divide the weight of the body 
in air by the difference between the weights in air and water. 

Example. 

A lump of glass is found to weigh in air 577 grains; it is 
then suspended by a horse hair from the bottom of the scale 
pan, and immersed in a vessel of pure water, when it is found 
to weigh 399.4 grains. What is its specific gravity ? 

577.0 Then, as 177.6 : 1 :: 577 : sp. gravity. 

399.4 1 



177.6 the difference. 177.6)577.0(3.248, &c. 

5328 



4420 
3552 



8680 
7104 

15760 
14208 



1552 

Answer, Specific gravity of glass is 3.248. 

The Hydrometer is an instrument constructed for the 
especial purpose of ascertaining the Specific Gravities of 
Liquids. 



Hand Book of Calculations. 



2IJ 



TABLE OF SPECIFIC GRAVITIES. 



Metals. 



Iron, (cast; 7.207 

« (wrought) 7.688 

Steel (soft) 7.780 

" (tempered) 7.840 

Lead (cast) 11.400 

" (sheet) 11.407 

Brass (cast) 8.384 

" (wire drawn). . . , 8.544 

Copper (sheet) 8.767 

" (cast) 8.607 

Gold (cast) 19.238 

" (hammered) 19.361 



Gold (22 carats) 17.481 

" (20 " ) 15.709 

Silver (pure, cast) 10.474 

" (hammered) 10.511 

Mercury (60°) 13.580 

Pewter 7.248 

Tin 7.293 

Zinc (cast) 7.215 

Platinum 21.500 

Antimony.. 6.712 

Arsenic 5.763 

Bronze (gun metal) . . . 8.700 



Stones and Earth. 



Coal (Bituminous) .... 
" (Anthracite).... -j 

Charcoal 

Brick 

Clay 

Common Soil 

Emery 

Glass 

Ivory 

(irindstone 

Diamond 

Gypsum ^ 



1.256 
1.436 
1.640 
.441 
1.900 
1.930 
1.984 
4.000 
3.248 
1.822 
2.143 
3.521 
2.168 



Lime 2.720 

Granite 2.625 

Marble 2. 708 

Mica 2.800 

Millstone 2.484 

Nitre 1.900 

Porcelain 2.385 

Phosphorus 1.770 

Pumice Stone 915 

Salt 2.130 

Sand 1.800 

Slate 2.672 

Sulphur 2.033 



Woods. 



Ash 845 I Cherry 715 

Beech 852 Cork 240 

Birch 720 ! Elm 671 



218 



Hand Book of Calculations. 



TABLE OF SPECIFIC GRAVITIES. 
Woods. 



Oak 1.120 

Pine (yellow) 660 

" (white) 554 



Poplar 383 

Walnut 671 

Willow 585 



Liquids. 



Acid Sulphuric 1.851 

" Muriatic 1.200 

Spirits of Wine 917 

Alcohol 790 

Oil (turpentine) 870 

" (olive) 915 

" (whale) 923 



Oil (linseed) 932 

" (castor) .961 

Pure water 1.000 

Vinegar 1.080 

Milk 1.032 

Sea water 1.029 



To find the weight of a cubic foot of anything contained in 
these tables. 

EULE. 

Multiply 62.5 lbs. (the weight of a cubic foot o£ pure water) 
by the specific gravity of the given body. 

Example. 

What is the weight of a cubic foot of sea water ? 

62.5 lbs. 
1.029 sp. gravity. 



5625 
1250 
6250 



Answer, 64.3125 lbs. is the actual weight; but 64 
lbs. is taken in practice as the weight of a cubic foot of sea 
water. 



Hand Book of Calculations. 2ig 

SPECIFIC GRAVITY. 

Example. 

How meny cubic feet of pea water will weigh a ton ? 
Divide 2240 lbs. (1 ton) by 64 lbs. 

8)2-240 
64 ' 

' 8) 280 



Ans. 35 

XOTE. 

35 cubic feet of sea water is always accounted to be a ton, as 
in sea water ballast for steamers, and in calculating displace- 
ment of ships. 







Example. 








is 


the 


weight of a cubic foot of wrought 
62.5 lbs. 
7. 09 sp. gravity. 


iron 


? 






5625 
3T50 
4375 









Answer, 480.625 lbs. 
480 lbs. in practice is the weight of 1 cubic foot of wrought 
iron. 

Example. 

What is the average weight of a cubic foot of Bituminous 
coal ? 

1.256 sp. gravity. 
62.5 lbs. 



0280 
2512 
7538 

Answer, 78.5000 lbs. 



220 



Hand Book of Calculations. 



SPECIFIC GRAVITY. 



This 78.5 lbs. is the weight of a cubic foot in a solid block; 
but loose, as used for fuel, a cubic foot weighs about 49. 7 lbs. 
which is the average of 13 kinds. 



Example. 
What is the weight of a solid cast cylinder of copper, 4 inches 



diameter and 6 inches high 

8.607 sp. gravity. 
62.5 lbs. 



43035' 
17214 
51642 



537.9375 lbs. per cub. ft. 



Say 538 lbs. 



.7854 

16 diam. squared. 



47124 

7854 



12.5664 area of base. 
6 high 



75. 3984 cu. in. in volume 



Say 75.4 cubic inches. 



lbs. 



cub. in. cnb. in. 

Then, as 1728 : 75.4 :: 538 : Answer. 
538 



6032 
2262 
3770 



12)40565.2 



1728 «{ 12) 3380.433 



12) 281.702 



23.475 lbs. 
Answer, 23 J- lbs. nearly. 



Hand Book of Calculations. 22 1 



ELEMENTS OF ALGEBRA. 



Algebra is a mathematical science which teaches the art of 
making calculations by letters and signs instead of figures. 

The name comes from two Arabic words, algabron, reduction 
of parts to a whole. 

The letters and signs are called Symbols. 

Quantities in algebra are expressed by letters, or by a combi- 
nation of letters and figures; as a, b, c, 2x, 3y, 5z, etc. 

The first letters of the alphabet are used to express known 
quantities; the last letters, those which are unknown. 

The Letters employed have no fixed numerical value of them- 
selves. Any letter may represent any number, and the same 
letter may represent different numbers, but in each sum the 
same letter must always stand for the same amount. 

The operations to be performed are expressed by the same 
signs as in Arithmetic; thus + means Addition, — expresses 
Subtraction, and x stands for Multiplication. 

Thus a-\-b denotes the sum of a and b and is read a plus b; 
a—b means a less b; and axb shows that a and b are to be 
multiplied together. 

Multiplication is also denoted by a period between the factors 
as a.b. But the multiplication of letters is more commonly 
expressed by writing them together, the signs being omitted. 

Example. — 7 abc is the same as Ixaxbxc. 

The sign of Division is ^-, thus a-±b is read a divided by b; 
but this is also expressed g; the sign of Equality is two short 
horizontal lines, as a = b and is read a equals b. 

The Parenthesis ( ) or Vinculum , indicates that the 

included quantities are taken collectively or as one quantity. 

Example. — 3 {a+b) and 3a-\-b each denote that the sum of a 
and b is multiplied by 3. 



222 Hand Book of Calculations. 



The character . ■ . denotes hence, therefore. 

A coefficient is a number or letter prefixed to a quantity, to 
show hoiv many times the quantity is to be taken. Hence a 
coefficient is a multiplier or factor; thus in 5a, 5 is a numeral 
coefficient of a. 

When no numeral coefficient is expressed, 1 is always under- 
stood. Thus xy means Ixy. 



DEFINITION AND EXPLANATIONS. 

An algebraic operation is combining quantities according to 
the principles of algebra. 

A Theorem is a statement of a principle to be proved. 

A Problem is something proposed to be done, as a question 
to be solved. 

The Expression of Equality between two quantities is called 
an Equation. 

An Algebraic Expression is any quantity expressed in alge- 
braic language, as 3a, 5a — 7a, etc. 

The Terms of an algebraic expression are those parts which 
are connected by the signs + and — . 

Thus in a-\-b there are two terms; in x, y and z— a there are 
three. 

A Positive Quantity is one that is to be added and has the 
sign -(- prefixed to it, as 4« +3&. 

A Negative Quantity is one that is to be subtracted and has 
the sign — prefixed to it, as 4a — 3b. 

A Simple Quantity is a single letter, or several letters written 
together without the sign + or — , as a, ab, 3xy. 

A Compound Quantity is two or more simple quantities con- 
nected by the sign -|- or — , as 3a+4£, 2x—y. 

The Axioms in algebra are self-evident truths as exemplified 
on page 130. 



Hand Book of Calculations. 223 



ADVANTAGES OF ALGEBEA. 

In algebra numbers are expressed by the letters of the alpha- 
bet and the advantage of the substitution is that we are enabled 
to pursue our investigations without being embarassed by the 

3sity of performing arithmetical operations at every step. 
. Tims, if a given nnmber be represented by the letter a, we 
know that 2a will r present twice that number, and \a the half 
of that number, whatever the value of a maybe. In like man- 
ner it' a be taken from a there will be nothing left and this 
result will equally hold whether a be 5 or 7, or 1000, or any 
other nnmber whatever. 

By the aid of algebra, therefore, we are enabled to analyze 
and determine the abstract properties of numbers, and Ave are 
nabled to molve many questions that by simple arithme- 
tic' would either be difficult or impossible. 

A working engineer has but little practical use for a too 
extended acquaintance with algebra, as nearly all the algebraic 
rules have been transferred to ordinary arithmetical computa- 
tion, but as the algebraic system is so inwoven into the school 
and college course of instruction it is well for everyone to know 
Borne thing of the elements of the science, as a traveller in a 
_u country is benefited by having some of the common 
- like bread, water, and the names of coin, even if he can- 
not comprehend the whole language. 

ALGEBRAIC FORMS. 

Arithmeticians for very many years have made a study of the 

use of formulae (this is Latin for the word forms) in stating 

problems and rules; these forms are nearly all expressed in 

lie term.-, as the prescriptions written for eoniponnding 

by druggists an- written in a dead language, it follows that one 

of the caua - for 30 doing is the same, i. e., the mystifying of 

■.'.ho have been practically educated outside the schools. 

The positive advantage however to be derived from the use of 

raic formulae is thai it puts into a shorl space whai other- 

Miight necessitate the use of a Long verbal or written 

explanation. 



22<f Hand Book of Calculations. 

Another advantage is that the memory retains the form of 
the expression much easier and longer than the longer method 
of expression. 

It may be remarked that those engineers wno once become 
accustomed to the method of use* of formulae seldom abandon 
their use. 

Examples Explaining the Solving of Formulae. 

1. If x = a -j- ~b — c + d — f; what must be the value of 
x when a = 10, h = 7, c = 9, d = 4, and/= 6? 

First substitute the figures for the letters, thus: — 

x = 10 + 7 — 9 -|— 4 — G„ then proceed as in the 
Arithmetical part. 

x== 21 — 15 = 6 Answer. 

2. If# = 4# r + 2m — 7 n — p -{- 3 q; find the value of x 
when # = 5; m = 3; n = 6; p = 1; and q = 8. 

Here 4 g = 4 times 5 = 20; 2 m = twice 3 = 6; 7 w = 7 
times 6 = 42; and 3^ = 3 times 8 = 24; 

Hence, x = 20 + 6 — 42 — 1 + 24 
= 50-43 
= 7 Answer. 

3. If# = ia — \d -\-\ c — £■/; find the value of # when 
« = 10; d = 24; c = 25; and/=12. 

As « = 10, then £ # = 5; as d = 24, then i £? = 6; as £ — 
25, then § c = 5; and as/ = 12, then f/= 9. 
Hence, a; == 5 — 6 + 5 —9 
= 10-15 
— — 5 Answer. 

4. If x = c — (| — jo); find the value of x when c = 8 

s = 3-j- and p = \\ 

x — 8 - (^ — H); here 3J- is divided by 2 = If 

= 8-(lf-ll) 

= 8-i 

= 7f Answer. 



Hand Book of Calculations. 225 



FORMULA. 

5. x = a b + c d — e f\ where a = 2, b = 3, c — 4, d = 5, 
<> = 6, and/ = 7. 

When two or more letters are joined together without any 
sign, it always means that they are multiplied together, hence 
the above becomes 

2=2X3+4X5-6X7 
= 6 + 20 — 42 
= 20 — 42 
= — 16 Answer. 

6. x = 4 a b c — 5 c d; a = 2; 5 = 5; c — 3 and f/ = 4 

=4X2X5X3—5X3X4 

= 120 — 60 

= 60 Answer. 
We have seen in the Algebraic part that such a quantity 
as jj means that a is to be divided by b; hence, i£ a = 24 and 
b — 6, then g = V = 4. 

AB 

7. Then if ^=_ . w hat is the value of x when A — 6; 

B = 7; C — 10; and D = 16 ? 

6X7 42 ^ A 

•"16 — 10- 6 = ' Answer * 

T + ^ 04 

8. What is the value of 1 + - T^q when T = 82 and 

% — 38 ? 

82 + 38 — 6 4 _ 56 

1 + ~ 1000 " — 1 + 1000 = X + - 05(j 

= 1.056 Answer. 
D 3 — //'■ What is the value of this when D = 14; 

9. 12 g of — 12; and £ — 15? 

14' _ 12 2744—1728 , t 1016 12102 

12 15 = 12 16 =12x -l5-= i:> 

= 812.8. Answer. 

Note. 

Instead of dividing 1016 by 15, and multiplying the quotient 

by. 12. we \>n-\'<T to first multiply L016 by 1'4, and then divide 

ise if yon divide by 15 firsl .you get a repeating 



226 Hand Book of Calculations. 

FORMULA. 

decimal, whereas by dividing last you bring the answer out 
exact. The same is true for the following one also. 

; (F— *)-(T — t) 

10. L = p _ T - X .009 C; what is the value of L 

whenP = 120; T = 67; t = 

32 and C = 3375 ? 

(120 — 32) (67 — 32) 
L = * UQ ; _2 67 L X .009 X 3375 

88 X 35 93555 



X 30.375 = — gg- = 1765.1886, &c. 

11. What is the value of U in the following: — 

U = 1115 + .3 T — t\ if T = 320 and t = 120 ? 
= 1115 + .3 x 320 — 120 
= 1115 + 96.0 — 120 
= 1211 — 120 = 1091. Answer, 
Note here, that the 320 must be multiplied, by the .3 first; 
the result, 96, must then be added to 1115, and the 120 sub- 
tracted from their sum. 

Also note, that in (8) (10) and (11) there is a capital T and 
a small I; also in (9) a capital D and a small d. In such cases 
the capital letter represents the larger quantity, and the small 
letter the smaller quantity. 

FOKMULA FOR DeTERMING STRENGTH OF BOILERS. 

txT 

P = X 2 

D 

P = bursting pressure; t = thickness of plate; T = tensile 
strength of the iron or steel; D = diameter of shell. 

Suppose your boiler to be 48-inch diameter of shell, of J inch 
plate having a tensile strength of 60,000 pounds per square 
inch of cross section, the rupturing pressure would be 

.25x60,000 
P = X 2 = 625 pounds. 

48 

If the boiler is single riveted, it would be limited to i of the 
maximum strength, or 104.16 pounds. With double rivets and 
holes drilled instead of punched, a working pressure of 125 
pounds might be allowed. 



Hand Book of Calculations. 227 



FORMULAE. 

Simple Formula for Horse Power of Engines. 
P A T 



33,000 

P king the mean effective pressure in pounds per square inch, 
A the piston area in square inches, and T the piston travel in 
feet per minute. 

ARITHMETICAL NOTES. 
Mathematics embraces three departments, namely: (I) Arith- 
metic- : ('£) Geometry, ineludmg Trigonometry and Conij 
Sections; (3) Analysis, in which letters are used, including 
Algebra, Analytical Geometry and Calculus. 

The first of the nine Arabic characters are called digits, from 
the Latin word digitus, a finger, owing to the fact that the 
ancients reckoned by counting the fingers. 

Figures have two values, Simple and Local. The simple 
value of a figure is its value when standing in units' place. The 
local value of a figure is the value which arises from its location. 

A continued fraction is a fraction whose numerator is 1 and 
whose denominator is a whole number plus a fraction whose 
numerator is also 1 and whose denominator is a similar fraction, 
and so on. 

Example. 

13 1_ 
54 — 4+l_ 

There are six cases of reduction: 1st. Numbers to fractions. 
2d. Fractions to numbers. 3rd. To higher terms. 4th. To 
lower terms. 5th. Compound to simple. 6th. Complex to 
simple. 

A circulating decimal is a decimal in which a figure, or set 
of figures is constantly repeated in the same order; as .333-j- 
.727272. 

M' assures are of seven kinds: 1. Length. 2. Surface or area. 
3. Solidity or capacity. 4. Weight, or force of gravity. 5. 
Time. G. Angles. 7. Money, or value. 



22S Hand Book of Calculations. 



STRENGTH OF MATERIALS. 



This is a general expression for the measure of capacity of 
resistance, possessed by solid masses or pieces of various kinds, 
to any causes tending to produce in them a permanent and dis- 
abling change of form or positive fracture. 

As a matter of calculation its principal object is to determine 
the proper size and form of pieces which have to bear given 
loads, or on the other hand to determine the loads which can 
be safely applied to pieces whose dimensions and arrangement 
is already given. 

The materials used in construction are chiefly of four kinds. 
1. Timber, 

2. Rock, or natural stones, 

3. Brick, concrete, etc. (artificial stones). 
4. Metals, especially iron. 

All these resist fracture in whatever way, but the capability 
of resistance in a given case varies with chiefly the following: 
1, the nature of the material and its quality; 2, the shape and 
dimensions of the piece used; 3, the manner of support from 
other parts; 4, the lines and direction of the force tending to 
produce rupture. 

Materials of all kinds owe their strength to the action of these 
forces residing in and about the molecules of bodies (the mole- 
cular forces) but mainly to that one of these known as cohesion; 
certain modified results of cohesion, as toughness or tenacity, 
hardness, stiffness, and elasticity are also important elements 
and the strength is in the relation of the toughness and stiffness 
combined. 

A piece of iron or timber may be subjected to strain or frac- 
ture in four ways: 1, it may be stretched, pulled or torn asun- 
der, as a tie-rod or a steam boiler. This is called tensile strain 
or tension, and is a direct pull; resistance to this force is called 
tensile strength. 2, the iron or timber may be crushed in the 
direction of the length as in columns and truss beams. This is 
direct thrust, direct pressure or compression; and the resistance 



I land Book of Calculations. 229 

THE STRENGTH OF MATERIALS. 

to it. the crushing strength. An example of this is found in 
the force tending to collapse the flues of a steam boiler. 3, it 
may be bent or broken across by a force perpend icular or oblique 
to its length, as in common bei:ms and joists. This is trans- 
strain or flexion; resistance to it the transverse strength. 
4, It may be twisted or wrenched off, in a direction about its 
axis, as in case of shafting. This is torsion; resistance to it 
orsional strength. 

Let it be noted, that any bending or breaking pressure is a 
; its effect on the piece a strain; briefly, then, the 
strength of a solid piece or body is the total resistance it can 
oppose to .-train in that direction. 

Important principles relating to strength of Materials. 

A r<'d. rope or an) r body being pulled in the direction of its 
length, its cohesion can come into play only by reason of the 
opposite length being fixed; and the amount of cohesion 
excited is a reaction against the strain applied; up to the limit 
of strength the amount of cohesion is always exactly equal to 
the acting strain; at every moment the strain and reaction are 
equal throughout the whole length of the piece acted upon. 

Where weight does not (as it must in any hanging rope or 

come in to modify the result the piece must, when the 

limit of strength is exe< ed< d, always part or yield at its weakest 

portion; that the tensile strength can never exceed that of such 

weakest portion. 

Two fibres of like character equally stretched must exhibit 
double the strength of one Generalizing this result, we say 
that the tensile Strength of beams, rods, ropes, wires, etc., is, 
for each material, proportional to the area of the cross-section 
of tie piece osed. This is, accordingly, also termed the abso- 
lute strength. 

When allowance for modifying influences is made, the laws 
of tensile strength become safe guides in practice, though the 
<.r of different materials in yielding to tension may vary 
considers 



2jo Hand Book of Calculations, 

THE STRENGTH OF MATERIALS. 

Any material, under a considerable tensile strain, becomes 
slightly elongated, not returning when the strain is taken off. 
This result is expressed by saying that the body possesses exten- 
sibility. It is doubtful whether in all materials, or in most, a 
result of this kind can be often or indefinitely repeated. But 
over this, the body lengthens a little by every pull in conse- 
quence of its elasticity; and this effect is not permanent, at 
least its whole amount is not so; the piece shortens again, when 
the strain is removed, by quite or nearly the amount of this 
lengthening. 

If the body possesses that of ductility, when the limit of its 
extensibility and elasticity is reached, the particles upon the 
surface at the weakest point begin to slip upon each other; the 
body is by this action both permanently and sensibly lengthened 
or drawn out, and as this extension does not, as in wire-draw- 
ing proper, take place under circumstances favorable to increase 
of toughness, the strength is with the first yielding impaired; 
while, if the load be not then diminished, the yielding portion 
must be drawn rapidly smaller until it parts completely. Thus, 
for ductile materials, the load beyond, which permanent change 
must occur is the limit of strength. 

In metallic bars or links, timbers, &c, a considerable pro- 
portion of the actual strength is gained by means of the firm 
hold of the fibres laterally one upon another; as is proved by 
the fact that, of two ropes of like material and containing in 
their sections a like number of fibres, in one of which the 
fibres are twisted and in the other glued together, the strength 
of the latter is greater by at least one third. 

In the tables of strength which follow, the piece experimen- 
ted on is (unless otherwise specified) always one the transverse 
section of which presents an area of 1 square inch; and the 
limits of strength found, known by the loads required to secure 
fracture, are expressed in pounds weight avoirdupois. 



Hand Book of Calculations. 



231 



THE STRENGTH OF MATERIALS. 



1. — Metals. 







Limits 






Limits 




Materials. 


of tensile 
strength. 


Materials. 




of tensile 
strength. 


Steel 


, best tempered 




Iron, ship plates, aver- 






134,000- 


-153,000 


age 




44,000 


Steel 


, cast, maximum 
shear 


112,000 
118,000 


" cast 


■1 


14,000 
45,970 


(( 


blister 


104,000 


" cast, mean 


of 




a 


puddled 


67,2C0 


American 




31,800 


a 


plates, length- 




Copper, wire 




61,200 












wise 


96,300 


" wrought.. 




34,000 


a 


plates, breadth- 




" cast, Ameri- 






• 


73,700 
150,000 


can 




24,250 


a 


razor 


Platinum, wire . . . 




53,000 


Ircn, 


wire.... 73,000- 


-103,000 


Silver, cast 




40,000 


( ( 


best Swedish bar 


72,000 


Gold, cast. 




20.000 


a 


bar, mean by 
Barlow 




Tin, cast block . . . 




3,800 




56,560 


" Banca 




2,122 


a 


bar, inferior . . . 


30,000 


Zinc 




2,600 


a 


boiler plates, av- 




Bismuth 




2,900 




erage 


51,000 


Lead, wire 

" cast 




2,580 
1,800 



2. —Other Materials. 



38, plate 9,400 

" flint 4,200 

Hemp fibres, glued 9,200 

II emp fibres, twisted 

(rope) 6,400 

Manila rope 3,200 

Marble, different Bpe-j 9,000 

ciea ( 5,200 

lifferent spe- j 1,000 

ee I 350 

k, well burned 750 



Mortar, of 20 years 52 

Roman cement, to blue 

stone 77 

Wood, box . . . .14,000— 24,000 
oak ....10,000—25,000 

locust tree 20,100 

elm 13,200 

asfa 12,000 

fir 8,330 

cedar 4,880 



2j2 Hand Book of Calculations. 



THE STRENGTH OF MATERIALS. 
Kule for Estimating Tensile Strength. 

The strength, per square inch section, of any material being 
known, this becomes for such material the unit or coefficient 
of strength. 

That is, the strength of a piece of any other section is (ap- 
proximately, of course) found by multiplying the unit for that 
material by the number of square inches in the transverse sec- 
tion of the piece. 

Example. 

If a bar of iron 1 inch square is torn asunder by 60,000 lbs., 
what will be required to break a bar 3f inches square ? Now 
then : 

3. 75 14. 0625 square inches. 

3.75 60, 000 tensile strength of 1 inch. 



2000 lbs to 



1875 Ton 843.750,0000 

2625 

1125 421+Tons Answer. 

14.0625 square inches m section of bar. 

Fatigue of Metals. 

A matter of great practical interest is the weakening which 
materials undergo by repeated changes in their state of stress. 
It appears that in some if not all materials a limited amount of 
stress variation may be repeated time after time without appar- 
ent reduction in the strength of the piece; on the balance 
wheel of a watch for instance, tension and compression succeed 
each other for some 150 millions of times in a year, and the 
spring works for years without showing signs of deterioration. 
In such cases the stresses lie well within the elastic limits; on 
the other hand the toughest bar breaks after a small numbei of 
bendings to and fro when these pass the elastic limits. 



Hand Book of Calculations. 233 



THE STEAM BOILER. 



To whatever use beat is to be applied through the medium of 
steam, the apparatus for generating and retaining the steam is 
structed on the same general principles for all purposes and 
is popularly termed a boiler. 

The most common types of steam generators may be arranged 
under the following designations: 

1. The plain cylinder boiler. 

2. The cylinder-flue boiler. 

3. The cylinder-tubular boiler. 
4. The return-flue boiler. 
5. The return-tubular boiler. 
6. Water-tube boiler. 

7-. The locomotive boiler. 

8. The sectional boiler. 
The primary conditions which steam generators should fulfill 
are: 1. strength to sustain the internal pressures to which they 
may be subjected. 2. Durability. 3. Economy or efficiency in 
evaporating qualities. 4. Economy of construction in materials 
and workmanship. 5. Adaptation to the particular circum- 
stances of their use. 6*. To these conditions must be added 
safety, which depends on form, construction, strength and 
quality of materials as well as management. 

In many forms or classes of steam boilers, the steam gener 
ating apparatus is not complete until the boiler is setup in 
brick work, with an external furnace constructed for the com- 
bustion of the fuel, and external flues made for conducting the 
heated gases along the sides of the boiler; in others the boiler 
ady for use as it comes from the manufacturer, having 
within the external shell all these necessary arrangements for 
COmbtlBtion and diMUght. 

In all cases certain appliances are necessary, such as the feed 
pump with the necessary pipes and attendants, the safety valve, 
grate bars, etc., so that a complete steam-producing apparatus 
requires much more than the single vessel called the boiler. 



ZJd Hand Book of Calculations. 



HORSE POWER. 

Eeference is made to pages 171, 172 and 173 for matter relat- 
ing to horse power of the steam boiler, to which is now added 
the following rules and calculations : 

To find the horse potver of a cylinder boiler. 

Rule. 

Multiply f the circumference in inches by the length in 
inches and divide by 144. The result is the heating surface in 
square feet. Divide this by 10 for nominal horse power. 

Example. 

What is the horse power in a plain cylinder boiler 40 feet 
long by 3 feet in diameter ? 

The circumference of 3 feet = 113 inches, 
frds of this = 75.3 inches. 

Multiply by length, 40 X 12 in. = 480 " 

Now: 60240 

( 12)36144 divided by 3012 



144^ 



( 12)3012 361440 



251 sq. ft. =251-^10 = 25 T V H. P. Ans. 



To find the horse poicer of a vertical boiler. 

1. Multiply the circumference of the fire box by its height 
above the grate, all in inches. 

2. Multiply the combined circumference of all the tubes in 
inches by their length in inches. 

3. Add to these two sums the area of the lower tube sheet, 
less the combined area in all the tubes — also in square inches. 

4. Divide the whole sum by 144 to obtain the square feet of 
heating surface. 



Hand Book of Calculations. 2j$ 

HORSE POWER. 

To find the square feet of heating surface in a locomotive 
boilrr. 

Rule. 

1 . Multiply the length of the furnace plates by their height 
above the grates in inches. 

2. Multiply the width in inches by their height in inches. 

3. Multiply length of crown sheet in inches by its widtL in 
inches. 

4. Multiply the combined circumference of all the tubes in 
inches by their length in inches. 

5. From the sum of these four products subtract the area of 
all the tubes and the area of fire door. 

ft Divide by 144 to get square feet of heating surface. 

To find the horse power of a flue boiler. 

Rule. 

1. Multiply § the circumference of shell in inches by the 
length in inches. 

2. Multiply the combined circumference of the flues in 
inches by their length in inches. 

3. Divide the sum of the products by 144, the result will be 
the heating surface in square feet. 

4. Divide this by 12 to get nominal horse power. 

Example. 

What is the horse power of a boiler 42 inches in diameter 
and 30 feel long with two 12 inch flues ? 

Circumfi renee of 42"== 131. 9xf =87.9 
Length of boiler in inches 42 X 12 = 504 



3516 
4395 

Now 443016-*- 144 — 307.6 

Divide 307A by 12 — 25 II. P. 443016 



2j6 Hand Book of Calculations, 

STRENGTH OF BOILER. 

To find the strength of a boiler. 

Rule. 

Multiply the tensile strength of the plate in lbs., by the 
thickness, in decimals of an inch; divide by diameter of the 
boiler in inches and multiply the product by 2; the answer will 
be the bursting pressure. 

Example. 

If a boiler be 48 inches in diameter and made of \ inch steel 
having a tensile (or tearing) strength of 60,000 to the square 
inch, then, 

60,000 
thickness .25 



300000 
120000 



diam. 48") 15. 00000 



31 2 i 
multiply 2 



625 lbs. strength to the square inch. 

Note. 

In practice, if the boiler is single riveted, ^th only of the 
above would be allowed as " safe " or 104to 6 ¥ lbs. on the steam 
gauge — with double rows of rivets and rivet holes, drilled 
instead of punched, a working pressure of 125 lbs. would be 
allowed. 

Factok of Safety. 

This is the number which expresses the ratio of the strength 
of the boiler to the working strain. 

The safe pressure depends upon the bursting stress or tensile 
strength of the plates, their thickness and the diameter of the 
boiler. 



Hand Book of Calculations. 



231 



STRENGTH OF BOILERS. 



In ascertaining the press- 
ure which is carried on the 
flat surface of a boiler, choose 
I stays as A B C in Fig, 110. 

Measure from A to B, and 
from A to U. The product 
of these is the number of 
square inches held by one 
stay: then proceed as in the 
following rule and example: 

To find the pressure on a 

certain number of stay bolts 
on a boiler. 




Fig. 110. 



Rule. 



Find the number of square inches enclosed by four adjacent 
bolts and multiply this area by the pressure of the steam as 
indicated by the steam gauge. This will give the strain on one 
bolt. Multiply this by the number of bolts upon which it is 
desired t<> find the strain and the result will be the answer to 
the question. 

Example. 
Suppose that the bolts are 4 inches apart, and that the gauge 
is 140 lbs., what is the strain on one stay bolt and what is it on 
8 bol- 

4x4 = 16 
pressure 140 



Number of bolts 



•14. 
10 

^.•^40 
8 



17.920 lbs. 

Note. 
The pressure on the surface <1<h>> not include the space occu- 
|)hm| by tie- area of the bolt, ; : be absolutely accurate 

that must !><• dedu< 



2j8^ Hand Book of Calculations. 

WATER CAPACITY OF A BOILER. 

To find the water capacity of a horizontal tubular boiler of 
any size. 

Rule. 

1. Multiply f of the area of the head in inches by the length 
of the boiler in inches. 

2. Deduct the area of a single tube multiplied by the number 
in the boiler multiplied by the length in inches. 

3. Divide by 231 to reduce the answer to gallons. 

Example. 

How much water (i being steam space) will a boiler contain 
6 feet in diameter and 18 feet long, with 100 3 inch tubes. 
The area of 6 feet in inches =4071.5 
and f rds of this is 2714.3 

Multiply by length 18 by inches X 12 = 216 



100 3 inch tubes to be deducted. 162858 

No. ft. in. 27143 

Area . 7 X 100 X 18 X 12 54286 

7X12 = 84 

586288.8 



8400X18 = 15120.0 

231)571,1688 ~ ~ 

24,726 galls. Ans. 

To compute the grate bar surface of a boiler furnace. 

Rule. 

Multiply the length and the breadth in feet and the result 
will be sauare feet of grates. 

Example. 

What is the grate bar area of a furnace 4J feet wide and 5 
feet deep ? 

4.5 
5 



22.5 Ans.— 22i 



Hand Book of Calculations. 239 

STEAM SPACE OF A BOILER. 
To find steam space of a horizontal boiler of any size. 

KULE. 

1. Multiply £ the area of the end of the boiler in inches by 
the length of the boiler in inches; the answer will be in cubic 
inches. 

2. To reduce to cubic feet divide by 1728. 

Example. 
How much steam space is there in a boiler 5 feet in diameter 
and 18 feet long ? 

The area of 5 feet (per table) is 282704^3 = 9423 . 
The length of the boiler, 18 feet, made into inches = 216 



Now divide : 1728 f 12)2035368 



v 



56538 



{ 12)169614 9423 

18846 

[ 12)14134 



2035368 



1.177Hth cubic ft. Ans. 
Note. 

These results are not absolutely accurate owing to discarding 
small fractions and not allowing for thickness of iron, but the 
calculations are sufficiently near to suit all practical purposes. 



240 Hand Book of Calculations. 



THE SAFETY VALVE. 



The safety valve is a circular valve seated on the outside of 
the boiler and weighted to such an extent that when the press- 
ure of the steam exceeds a certain point the valve is lifted and 
allows the steam to escape. 

Safety valves can be loaded directly with weights, in which 
case they are called dead weight valves, or the load can be trans- 
mitted to the valve by a lever. An unauthorized addition of a 
few pounds to the weight of the former would make no appre- 
ciable addition to the blowing oif pressure while a small addition 
to the weight at the end of the lever is multiplied several times 
at the valve. 

In the case of locomotive and marine boilers the lever is 
weighted by means of a spring, the tension of which can be 
adjusted. 

It may be denned, as also applying to all valves, that the 
seat of the valve is the fixed surface on which it rests or against 
which it presses, and the face of a valve is that part of the surface 
which comes in contact with the seat. The spindle is the small 
rod which projects upwards or downward from the middle of 
the valve, and so arranged that it causes the v r alve to raise and 
drop evenly upon its seat. 

The effective pressure on the lever safety valve can be regu- 
lated within certain limits by sliding the weight along the arm 
and in the spring safety valve the pressure can be regulated by 
altering the tension of the spring. 

Every boiler should be provided with two safety valves. The 
size of the opening into the boiler depends upon its steam pro- 
ducing qualities, the object to be attained being to reduce the 
pressure within the boiler to its safety point as quickly as pos- 
sible. 

Safety Valve Calculations. 
This is a subject, while old, is ever new to the engineer, and 
the following rules are given in such a manner that any one who 
can add, subtract, divide and multiply and read decimals can 



Haul Book of Calculations* 



241 



LEVER SAFETY VALVE. 

soon acquire familiarity with the rules, and thus make them 
his own to use when he needs them, to adjust a safety valve or 
for other uses. 

Reference is first made to the principles, rules and examples 
heretofore given relating to the lever (the safety valve is a lever 
of the third order) the rule of three,, and to decimals. Next, in 
all problems it is well for the engineer to draw, roughly, if need 
be, a diagram of a safety valve somewhat after the form given 
in Fig. 111. 



•<--->--: 





Fig. 111. 



W denotes the weight on the lever in pounds; L, distance 
from center of weight t<> fulcrum in inches; w, weight of the 

itself in pounds; g, distance between center of gravity of 
lever and fulcrum In Inches; I, the distance between center of 

and fulcrum in inches, and F the fulcrum. 

In working out the problems the figures and dimensions as 

soon as known should be pnl upon the drawing bo thai the eye 

in the calculations as the) proceed from step to step. 



2^.2 Hit iid Book of Calculations. 

LEVER SAFETY VALVE. 

To find the weight of the valve, spindle, lever, etc., proceed as 
follows : 

Take out the valve and spindle and weigh them and make a 
note of it, then put them back in place, connect the lever and 
drop it in place resting on the valve spindle, tie a string to the 
lever directly over the spindle, hook on the scales to the string 
and weigh the lever, to the weight of the lever add the weight 
of valve end spindle, or the weight may be found approximate- 
ly by computation, by use of rules elsewhere given in this wor& 
under Mensuration, etc. 

The following rules were recently issued by the United States 
board of supervising inspectors, on account of changes in the 
rules for granting licenses to engineers of steam vessels. 

To find the weight required to load a given safety-valve to blow 
at any specified pressure. 

1. Measure the diameter of the valve, if it is not known, 
and from this compute its area exposed to pressure. 

2. Weigh the valve and its spindle. If it is not possible to 
do this, compute their weight from their dimensions as accu- 
rately as possible. 

3. Weigh the lever, or compute its weight from its dimen- 
sions. 

4. Ascertain the position of the centre of gravity of the lever 
by balancing it over a knife-edge, or some sharp-cornered 
article, and measuring the distance from the balancing point 
to the fulcrum. 

5. Measure the distance from the center of the valve to the 
fulcrum. 

6. Measure the distance from the fulcrum to the center of 
the weight. 

Then compute the required weight as follows: 

1. Multiply the pressure in pounds per square inch at which 

the valve is to be set by the area of the valve in square inches; 

set the product aside and designate it " quantity ~No. 1." 



Hand I ■ Calculations. 24.3 



THE LEVER SAFETY VALVE. 

2. Multiply the weight of the lever in pounds by the distance 
in inches of its center of gravity from the fulcrum; divide the 
product by the distance in inches from the center of the valve 
to the fulcrum, and add to the quotient the weight of the ralve 
and spindle in pounds; sum aside and designate it 
" quantity No. v.*' 

3. Divide the distance in inches from the center of the valve 

.11 1 y the distance, also expressed in inches, from 
the center of the weight to the fulcrum; designate the quotient 
"quantity No. 3." 

4. Subtract quantity No. 2 from No. 1, and multiply the 
difference by No. 3. The product will be the required weight 
in pound-. 

To find the ?< m th of the lever, or distance from the fulcrum 
at which a gi cjht must be set to cause the valve to blow at 

any > \ ire. 

The ar< a 1 t the valve in square inches, the weight of the 
valve. Bpindle and lever in pounds, the position of the center of 
gravity of the lever, and the distance from the center of the 
vahe to the fulcrum, must be known, as in the first example. 

Then compute the required length as follows: 

1. Multiply the area of the valve in square inches by the 
in pounds per square inch at which it is required to 
blow; s t the product aside, and designate it "No. 1." 

1. Multiply the weight of the lever in pounds by the distance 
in inch< ity fr< m the fulcrum; divide the 

pr< duct by the distance in inches from the center of the valve 
to the fulcrum; add to the quotient the weight of the valve and 
spindle; s< aide, and designate it " No. \\" 

:;. Divide the distance in inches from the center of vahe to 
fulcrum by I s;ht of the ball in pounds, and call the quo- 

tienl 

L. Subtrs • "No. 2" from "No. l." and multiply the 
irence by " No. 3"; the product will express the distance 
in inch*- that the ball musl be placed from the fulcrum 
produce I be required pressure. 



244 Hand Book of Calculations. 

THE LEVER SAFETY VALVE. 

To find at what pressure a safety valve will commence to Now 
when the weight and its position on the lever are known. 

The weight of valve, lever, position of centre of gravity of 
lever, etc., must be known as in both the preceding examples. 

Then compute the pressure at which the valve will blow, as 
follows: 

Multiply the weight of the lever by the distance of its center 
of gravity from the fulcrum; add to this product that obtained 
by multiplying the weight of the ball by its distance from the 
fulcrum; divide the sum of these two products by the distance 
from the center of the valve to the fulcrum, and add to the 
quotient so obtained the weight of the valve and spindle. 
Divide this sum by the area of the valve; the quotient will be 
the required blowing-off pressure in pounds per square inch. 

Example. 

Suppose we have a safety valve, with a weight of 50 lbs. sus- 
pended 24 inches from the fulcrum; say the lever weighs 6 lbs., 
gravity center (balancing point) 15 inches from the fulcrum, 
weight of valve and spindle 2 lbs., and its center 4 inches from 
the fulcrum, and the diameter of the valve 2 inches, at what 
pressure will the valve open ? Now then: — 

Diameter of valve is 2 inches; its square is 2x2= 4; its area 
is 0.7854x4 = 3.1416; the weight of the ball is 50 lbs., its dis- 
tance from fulcrum is 24 inches, and 50x24 = 1,200; the 
weight of lever is 6 lbs., the center of gravity is 15 inches from 
the fulcrum, and 15x6 = 90; the weight of the valve is 2 lbs., 
and its distance is 4 inches from fulcrum, and 4x2=8; the 
area of the valve is 3.1 41 H, and its center is 4 inches from ful- 
crum, then 4 X 3.14.16 = 12,5664, and 1200+90+8= 1298, and 
1298 divided by 12.5664 = 103.03 lbs., or the pressure at which 
the valve will open. 

Note. 

The " moment " or leverage of the steam is the total pressure 
acting apwards, multiplied by the distance in inches from the 
pivot to the valve-stem. The moment or leverage of the ball 
acting downwards is the total weight of the ball multiplied by 
the distance in inches from the pivot to the center support of 
the ball. 

"When therefore the moment of the steam, which acts upwards, 
exceeds both the dead weight of the lever and valve, and also 
the moment of the ball holding the valve down, then the valve 
rises and steam escapes. 



Hand Book of Calculations. 



2 15 



SIMPLE RULES RELATING TO THE SAFETY VALVE. 

« 

N>>\v the weight of the lever and valve is of so little import- 
ance m the matter of pressure, that working engineers usually 
omit it from their calculations, which may wisely be done, as 
the simplest rules arc generally the best for engineers. 




Fig. 11! 



In the third form of the lever the power is at one end, the 
fulcrum at the other, and the resistance to be overcome some- 
wheiv between them. 

Rule 1. Multiply the length of the lever by the power and 
divide the product by the short arm; the quotient is the resist- 
overcome. 

Example. 
The length of the lever is 10 inches, with a power equal to 
80 1!)-.. acting at one en 1; what resistance will this power over- 
coin.-, th.' short arm being 6 inch' 

Length of the lever, 10 inches. 

Multiplied by the power, 30 1b.--. 

-iiort arm, 6 480 

A; ristance. 



2J/.6 Hand Book of Calculations. 

THE LEVER SAFETY VALVE. 

The resistance, the short arm, and the poiver given, to find 
the length of the lever. 

Rule 2. Multiply the resistance by the short arm, and divide 
the product by the power; the quotient is the length of the 
lever. 

Example. 

The short arm is one inch, the resistance to be overcome is 12 
lbs., the power to be applied is 2 lbs.; required the length of 
the lever. 

The resistance, 12 lbs. 

Multiplied by short arm, 1 inch. 

Divided by the power, 2 lbs.)12 

Answer 6 inches length of lever. 

The resistance, the short arm, and the length of the lever to 
find the power. 

Rule 3. Multiply the resistance by the short arm, and divide 
the product by the length of the lever; the quotient is the 
power required. 

Example. 

If you wish to weight the safety valve of a steam engine, 40 
lbs., with a lever 10 inches long, what power must you apply, 
allowing the short arm to be two inches. 

The resistance, 40 lbs. 

Multiplied by the short arm, 2 inches. 

Divided by the lever, 1 0)80 

Answer, 8 lbs. the power. 

The power, length of ihe lever, and the resistance given, to 
find the short arm. 

Rule 4. Multiply the length of the lever by the power, and 
divide the product by the resistance; the quotient is the length 
of the short arm. 



Hand Book of Calculations. 2J.J 



THE LEVER SAFETY VALVE. 

Example. 
The length of the lever is 5 inches, with a power equal to 3 
lbs. applied: the resistance to be overcome is 20 lbs.; required 
the length of the short arm. 
L i LLtli of lever, 5 inches. 

Multiplie I by the power, 3 lbs. 

Divided by resistance, 20 lbs.)15 

Answer, . 75 inches, the length of short arm. 
These rules can be learned easily, and will be found very 
useful for engineers. They comprise all the problems incident 
fcting a lever safety valve. 

Other Rules Relating to the Safety Valve. 
Rule for finding the pressure to raise the valve and ball. 
Divide the lever (in inches) by the short arm (in inches), 
divide the weight of ball by the area of valve and multiply the 
two quotients together. 

Example. 
Area of valve 124- inches, length of lever 25 inches, length of 
short arm 2 inches, weight of ball 75 pounds, 75-=-12-£ = 6, 
25-j-2 = 12£ X G = 75 pounds pressure per square inch to raise 
the lever. 

11 nh- to find zchere to place the iveight on the lever to allow the 
ralre to open at a desired pressure. 

Multiply the weight of lever by the horizontal distance of its 
center of gravity from fulcrum. (2) The weight of valve by 
distance from fulcrum. (3) Area of valve by steam pressure 
per square inch and by distance from fulcrum. Add together 
the iir.-t two products, subtract their sum from the third, and 
divide the difference by the weight of the ball. 

To Test Correctness of Calculations 

The proper way to set the weight on a Bafetj valve lever to 

the figuring i- to raise steam on the boiler to the pressure 

red, as marked on a correct steam gauge, and adjust the 

weight .-'(that tin- valve just "simmers/' Whena safety valve 

w.:. • by " calculation, " the valve never blows off at the 

g often some pounds out of truth, owing 

to f rid l the parts'. 



248 Hand Book of Calculations. 

CHIMNEYS. 

A chimney promotes a flow of air through a furnace, because 
the hot air contained in the chimney is lighter than the sur- 
rounding atmosphere, which consequently endeavors to force 
i -8 way into the chimney from below in order to restore the 
balance of pressure. The only way into the chimney is through 
the fire-bars and furnace, and in passing' through these the air 
maintains the combustion, and at the same time becoming 
itself heated, makes the action of the chimney continuous. 

. In estimating the action of a chimney of a given size in pro- 
ducing a draught, the density, temperature, and volume of the 
products of combustion must be considered. 

The draught increases directly as the area and as the square 
root of the height. If either is assumed or determined upon 
the other may be found from the formulae, 
120 X grate surface in sq. ft. 

=area in sq. inches. 

*J height in feet 

Example. 
What should be the size (area in sq. inches) of a 100 feet 
chimney with grates 5-J feet deep by 10 feet wide. Now then: 

120X5^X10=6,600 



divide by 10 (the sq. root of 100) 660 sq. in. 
660 sq. in. =26 in. square, nearly. 

To find the number of cubic feet of air in a chimney. 

Eule. 
Multiply the length in feet by area in feet or decimals of a 
foot and the answer will be the contents of air in cubic feet. 

Example. 
How much air will be contained in a chimney 90 feet high 
and 48 in. square flue ? 

4x4=16 area in feet X 90 = 1440 cubic feet Ans. 
Example. 
If it be an iron chimney 90 feet high and 24 inches in diam- 
eter ? — then 

Area 24 in. (per table) =452. 4 sq. in. =3.141 sq. feet. 

90 



282 T 7 oths Ans. 



Hand Book of Calculations. 24.9 

THE STEAM ENGINE. 

A steam engine may be defined as an apparatus for doing 
w«»rk by means of heat applied to water. The complete study 
of the steam engine involves an acquaintance with the sciences 
of pure and applied mechanics: of chemistry; of heat; as well 
knowledge of the theory of construction, and the strength 
of materials. 

The steam engine as it exists to-day is the growth of two cen- 
turies of experiments and practical application of the mechan- 
ism to all sorts of work; the history of its successive steps of 
advancement, while fall of interest, is too voluminous for this 
work. 

In the future as well as in the past, questions relating to the 
influence on steam economy, of speed in the engine, of pressure 
and ratio of expansion of steam, or of superheating, must in 
the main, be settled by an appeal to experiment, — experiment, 
guided and interpreted by the great underlying principles of 
thermo-dvnamics and the theory of steam, outlined in preced- 
ing pages of this work. 

In the steam engine, heat accomplishes work only by being 
let down from a higher to a lower temperature, A certain 
amount of heat disappears when changed into work. 

Classification and Varieties of Engines. 

The stationary engine is the most perfect form of the steam 
engine. In this type, economy of steam is carried to its high- 
in the locomotive, the steam fire engine and other 
portable engines, and still more so, in the steam hammer, etc., 
it is i;n.' ssible to apply many of those means by which steam, 
and consequently fuel, is economized. 

re of two kinds, called respectively low 
pressure and high pressure engines. These terms do not refer 
to the initial -team in the cylinder but to its final 

. The terms are, however, inappropriate, since they do 
Dot express the distinctive difference between the two varieties 
diff< rence may be briefly expressed as follows: 
_m pressure discharges its steam directly into the 

atmosphere; and lentlythe steam on leaving the cylin- 

der • force 1 qua) to, al Least, 15 lbs. to the 

square inch. The wfc rce is of course wasted. 



2^0 Hand Book of Calczdations. 

THE STEAM ENGINE. 

The low pressure engine condenses its steam and discharges 
it as water; and consequently the pressure of the steam on 
leaving the cylinder may be much less than that of the atmos- 
phere. Thus a large proportion of the steam wasted by the 
high pressure engine is utilized. The terms condensing and 
non-condenslkg engines are therefore much more appropri- 
ate than low and high pressure. 

The difference in effect between the condensing and non-con- 
densing engines, with equal pressure of steam and expansion, is 
solely that the condensing engine has the advantage of the 
effect produced by the vacuum or amount of atmospheric 
atmosphere removed, which varies according to the perfection 
of the machinery, from 10 to 13 lbs. per square inch; some of 
the waste heat however in the non-condensing engine is utilized 
by leading the exhaust steam through a heater. 
Simple and Compound Engines. 

There is another classification of engines into simple and 
compound, the latter being those in which steam is used twice 
by being exhausted from one cylinder into another, while the 
former applies to all engines which use steam only once, whether 
they are double engines and have double sets of valve gear or 
not. Locomotives, steam fire engines and stationary engines 
which take their steam directly from the boiler and exhaust it 
into the atmosphere should be termed simple engines, regardless 
of the number of cylinders; hence the term single engine some- 
times used is incorrect. 

In the compound engine the steam is first admitted into the 
small or high pressure cylinder until the piston has moved 
through a certain distance, when the valve is so regulated that 
the communication with the boiler is cut off, the remainder of 
the space to be passed through by the piston being performed 
by the exj)ansion of the steam, which, having done its work, 
escapes to the second, or condensing cylinder, where it does a 
proportionate amount of work and out of which it escapes into 
the condenser. 

The receiver is a chamber between the cylinders of compound 
engines into which the steam irom the high pressure cylinder 
escapes and from which it is admitted to the low pressure cyl- 
inder. 



Hand Book of Calculations. 2$r 



MARINE ENGINES. 

A marine engine properly speaking is an engine designed to 
occupy a certain space in a vessel and to furnish a certain 
amount of power; hence the marine engine may be either con- 
densing or non -condensing, vertical, horizontal or inclined, 
simple or compound. The most desirable class of marine 
engines are those that develop the greatest amount of p:>\ver 
with a given area of piston and steam pressure, and that occupy 
the hast spaa . 

The pressure now commonly used at sea with improved types 

steam engines may lie said to vary from about 80 lbs. to 160 
lbs. per square inch, and the consumption of coal from 1^ lbs. 
: lbs. per indicated horse power per hour. 

To withstand such pressure, the shell plating of such boiler 
i.< made of a thickness of 1 inch or more; the end plates are 
usually about 1-t per cent, thicker. 

Generally the total heating surface in marine boilers is from 
25 to 28 times the grate area and the tube surface is about §ths 
of this. 

The tubes are about 6 feet long and 3 inches in diameter. 

The heating surface is sometimes stated as varying from 16 
o square feet per nominal horse power, the indicated horse 
power being from 5 to 0" times the nominal horsepower; or the 
heating surface may be stated as about 3 square feet per indi- 
d horse power, the grate area being about ?Vth of the heat- 
in g surface, or from {th to yoth of a square foot per indicated 
power. 

The Rotary Engine. 

Each of the types of engines named are still further divided 

into numerous varieties, often blending into each other and 

combining the essential principles of two or more systems, but 

all acting u;> n the same general law of pressure and expansion 

combined with the velocity of the piston regulated by the 

rnor. From this general statement must be 

Laded tie- rotary engine, which works npon the sole pioperty 

scaping steam. From the days of James 

d an expi nmeni and qcv< r reduced to 



252 



Hci7id Book of Calculations. 



THE MECHANISM OF STEAM ENGINES. 
The variety of form and arrangement of the parts of steam 
engines is so great that an extended knowledge of the mechan- 
ism of its -various parts can only be gained by close observation 
of numerous engines, or working drawings. 

The Cylikdee. 
The cylinder of a steam engine is the closed vessel in which 
the piston works backwards and forwards. It is so called 
because the interior is cylindrical in shape, though the form of 
the exterior is complicated by sundry additions. It is made of 
cast iron, the interior being carefully bored so as to form a 
smooth round surface for the passage of the piston. It consists 




Fig. 113. 
of the following parts. The cylindrical body AA, which is cast 
in one piece; — the steam chest BB, in the thicknesses of which 
are formed the two steam passages SS and the exhaust passage 
E; — the two covers CO, which are flanged, and which are 
attached to the body of the cylinder by means of studs and nuts. 
The cover through which the piston rod works is provided with 
a stuffing box D and gland e, to prevent the steam escaping 
round the rod. This object is accomplished in the following 
manner. The space aa between the rod and the inner cylindri- 
cal surface of the stuffing box is filled with plaited hemp satu- 
rated with tallow, or with one of the numerous patent packings 
now procurable. The gun -metal gland e, through which the 
piston rod passes, is forced up against the packing by means of 
the *t wo nuts and screwed studs shown in the drawing. The 



Hand Book of Calculations. 253 

THE STEAM CYLINDER. 

result is that the packing can be squeezed with any desired 
degree of tightness round the piston rod, and can thus prevent 
the escape of steam. The opening by which the piston rod 
passes through the substance of the cover is lined with a gun- 
metal bush, c. In many engines there is a stuffing box on the 
other cylinder cover, through which a prolongation of the 
piston rod works. The object of this arrangement is to pre- 
vent the piston bearing unequally on the lower side of the 
cylinder. It is also adopted in the case of condensing engines, 
when the plunger of the air-pump is driven direct from the 
piston. It will be noticed that the interior faces of the covers 
are - to fit into the corresponding faces of the piston. 

The reason cf this arrangement is, if the cover faces were not 
ped correspondingly, there would be at each end of the 
space to be tilled with steam before the piston 
_ di to mow, which steam would do no work till expansion 
_ .m. 

Owing to 1, the invisibility of steam, 2 the complications of 

the parts, and 3, inattention and thoughtlessness, very many 

ra in charge, to say nothing of their assistants, do not 

know the method of entrance and exit of steam from the cylin- 

3 of their engines, and yet this is one of the first tilings to 

be learned. 

capacity of a steam cylinder 
Rule. 
tiply the length in inches by the area in inches; the 
- in cubic inch 

Kx AMPLE. 

What is the total capacity of a 15£x30 inch .-team cylinder? 
Area= I - e Table page 116) 

30 



f 12)5660.7570 



1;--- !•< L7.17297 



[ r. 



1 — An-, in cubic 



254 Hand Book of Calculations. 



THE STEAM CYLINDER. 

To find the diameter of a cylinder for any given horsepower. 

Rule. 
Multiply horse power desired by 33,000, divide by piston 
speed in feet per minute and again by mean pressure in cylin- 
der and the result will be the area of piston, which divided by 
.7854 will give the square of the diameter. 

Example. 
75 horse power is desired from an engine running 120 turns 
per minute with cylinder 15 inches long, mean pressure 80 lbs. 

75 
33,000 

2^5000 
225 



piston feet 300)24750.00 



pressure 80)8250.00 



.7854)1031.2500(131.3 =sq. of diam. 
The square root is 7854 
131. 3(11. U nearly 

! 24585 



2356 



21) 31 



21 10230 

7854 



224)1030 



896 23760 

23562 

Answer llij- inches nearly. 
Definitions. 
1. Mass. This word denotes the quantity of matter con- 
tained in a body. 2. Weight is the attraction which the earth 
exercises on a mass. 3. Velocity is the speed at which a body 
moves; i. e. the space which it traverses in a given time. 
Motion. This word is employed in mathematics to denote 
movement on the part of a body and also takes account of the 
mass of the body moved — hence the terms momentum and 
moment. .Force is any cause which produces the motion in a 
bod v. 



Hand Book of Calculations. 255 



THE STEAM CYLINDER. 
To find the area of a piston, rilher water or steam. 

Rule. 
Square the diameter and multiply by .7854. 

Example. 
What is the area of a piston 4+ inches in diameter ? 
4i squared = 20.25 
.TS54 

8100 
10125 
16200 
14175 



15.004350 Ans. 15 r 9 oths. 
Note. 

The use of the Tables of Diameters,, etc., pages 114-126 may 
be illustrated in this problem. On page 114 the diameter 4.5 
being given, the area is carried out 15.9043= the above. In 
the next line 14.1372=£Ae circumference of the same 44- inches. 

Pistons. 

The piston is the metallic disc which accurately fits the bore 
of the cylinder, and which receives and transmits the pressure 
of the steam to the other moving parts of the engine. 

Tlu- forms of pistons are innumerable, and depend altogether 

upon the purpose for which the engme is intended, and the 

size of the cylinder, which in different classes of engines varies 

from a few inches to ov< r 9 feet in diameter. The chief points 

end to in the design of .1 pi-ton are the following:, it 

should be strong enough towithscand the pressure of the steam, 

and to hold the end of the piston rod immovably; — the packing 

round the circumference should be steam-tight, without can* 

ttdue friction, and nol liable to gel out of order; — the 

width of the circumferential portion should be such thai the 

re per square inch— due, In I he case of horizontal engines, 

to the weighl of the piston— be not sufficient to cause undue 

tnni ]• surface of the 1 \ lindi 



2 5 6 



Hand Book of Calculations. 



THE STEAM CYLINDER. 

The importance of keeping the surface of the cylinder true, 
and of keeping the piston in steam-tight contact with it, will 
be readily recognized when it is borne in mind that a leakage 
of steam past the piston means that during the whole time the 
engine is at work there is an open passage from the boiler to 
the condenser or outer air, through which steam is continuously 
escaping without doing any work. 




. Fig. 114. 
In the case of many engines, the packing ring is usually 
pressed against the barrel of the cylinder by means of a series 
of independent adjustable springs contained within the body of 
the piston. The spring ring, which is of considerable depth, 
is held up against the sides of the cylinder by a series of steel 
springs, a a a, Fig. 114. The joint in the ring is formed to pre- 
vent leakage. An oblique slot is taken out of the ring. A 
plate, fitted with a tongue piece, is fastened behind the slot, 
and the tongue piece, which slides in a groove, allows the ring 
to expand and contract, and at the same time makes a steam- 
tight joint. The ring, with its springs, is covered by a flat 
circular piece of iron called the junk rmg, which is shown in 
plan on one half of Fig. 114. This enables the springs to be 
got at easily for examination and repair. The junk ring is 
attached to the body of the piston by bolts which work into 
brass nuts embedded in the metal of the piston. When the 
threads work loose, the nuts can be easily replaced. 



Hand Book of Calculations. 



2 57 



Piston Eods. 
The piston rod is the member which transmits the motion 
imparted to the piston to the mechanism outside the cylinder. 
It consists of a truly cylindrical bar of wrought iron or steel, 
one end of which is fastened securely into the piston. The 
rod passes through the cylinder cover by means of a steam-tight 
stuffing box, as shown in Fig. 113, and the outer end terminates 
in the cross-head. There are various methods in vogue of 
fastening the rod into the body of the piston. Sometimes the 
end of the rod is turned cylindrical and a hole bored in the 
piston slightly less in diameter than the rod. The piston is 
then heated which causes it to expand, when the rod can be 
inserted. After cooling, the piston contracts, and holds the 
rod firmly in its place. In the majority of cases the end of the 
rod is turned conical as in Fig. 115, with a screw thread on the 




I 



Fig. 115. 

extreme end, by means of which, together with a nut, the rod 
is firmly embedded in a conical recess bored in the piston. 

The strength of piston rods has to be fixed with special refer- 
ence to the fact that they are subject to alternating strains. 
When the piston is making the stroke towards the crank shaft 
tin- rod is in compression, and when making the return stroke 
the rod 13 in tension. The maximum stress per square inch 
of cross section at <<iiij part of the stroke is equal to the total 
pressure of the steam on flic piston divided by the area of the 
rod. It is usual in designing pieces of machinery which have 
to bear alternating strains of tension and compression to make 
them much stronger than would be necessary were the strain 
ahvaws of one sort. 



*5* 



Hand Book of Calculations. 



Connecting Rods and Cranks. 

The connecting rod is the link which enables the back and 
forth motion of the piston to be converted into the circular 
motion of the crank pin. It is a link or rod of metal so formed 
at the two ends that it can be jointed to both the cross-head 
and the crank -pin. 

Fig. II G represents a connecting rod, and its various details 
as used in a stationary engine. The parts surrounding the 




Fig. 116. 
crank and cross-head pins are made of gun metal, brass, or 
white metal so as to diminish friction. They are made in 
separate pieces, called steps, and are held in place by the straps 
aa, which are fastened to the rods by means of the gibs M, and 
the cotters cc. When the brasses wear they can be tightened 
up by driving in or screwing up the cotter, which draws up the 
strap, and thus tends to shorten the rod. 

Cross-heads and Slide-baks. 

The outer end of the piston rod is attached to the cross-head, 
or motion block, which serves the double purpose of forming 
the means of connection between the piston rod and the con- 
necting rod, and of guiding the piston rod so as to keep it 
straight and in the line of the axis of the cylinder, in spite of 
the bending moment due to the angular position of the con- 
necting rod. 

The cross-head generally consists of three principal parts, 
viz. (1) the body which often contains a conical hole into which 
the coned end of the piston rod is fastened; (2) the part by 



Hand Book of Calculations. 



2 59 



CROSS-HEADS AND SIDE-BARS. 




Fig. ir 



which the joint with the connecting rod is 
effected; and (3) the guides or motion blocks 
which travel between fixed bars parallel to the 
axis of the cylinder, called slide-bars, and which 
prevent the end of the piston rod from being 
deflected as the connecting rod assumes an angu- 
lar position. Fig. 117 shows a piston rod, 
cross-head, slide-bars, and connecting rod in 
position. 

Cranks and Eccentrics. 

The crank is simply a lever of the first order, 
either attached to, or forged in one piece with 
the main shaft of the engine. By means of 
it, the reciprocating motion of the piston is 
finally converted into circular motion. 

Fig. 118 shows two views of one of the sim- 
plest forms of crank; A is the crank shaft, c 
the crank pin. The distance from the center 
of A to the center of c is the length of the 




c 


l 


\ 

) 



A = crank shaft ; C = crank pin ; B - web; 
D, D' = bosses; E = key 

Fig. 118. 



crank arm, which is, of course, equal to half the stroke of the 

Ion. B is the web of the crank, D, D' the bosses. Cranks of 

this form are generally of cast iron, and are attached to the 

main shaft by means of a key K. The fastening of a movable 



260 



Hand Book of Calculations. 



CRANKS AND ECCENTRICS. 

crank on a shaft requires the greatest care, because all the 
stresses thrown on the crank are liable to reversion during each 
stroke, especially in the case of a slow-running engine working 
with a considerable cushion of steam. Very severe reversals of 
pressure also occur if water is allowed to accumulate in the 
cylinder. In such cases the piston is brought up dead before 
the end of the stroke is reached, while the crank endeavors to 
pull on, thus throwing a heavy strain on all the connections, 
and amongst others on the key. Cranks of this type in addi- 
tion to being keyed are generally shrunk on to the shaft, or 
else are forced on by hydraulic pressure. 




Fig. 119. 



Pig. 119 shows an end elevation and cross-section of another 
form of cast-iron crank, called a disc crank. It is, as its name 
implies, formed of a disc of cast iron, attached to the shaft by 
the methods just described, and provided with a wrought iron or 
steel crank-pin. The portion of the disc opposite to the pin is 
usually much thicker and heavier than the remainder of the 
disc, this extra weight being used as a balance to the weights 
of the reciprocating parts. 

The crank-pin is the portion of the 'engine which receives the 
greatest stress, and special care must therefore be given to its 
design and lubrication. See Cut 122. 



Hand Book of Calculations. 261 

The Eccentric. 

There is a species of crank called the eccentric, in which the 
pin is so large that it completely envelopes the shaft. Such a 
crank is shown at Fig. 121 (B). The distance ac is the same 




Fig. 120. 

as at A, but the pin has assumed the diameter of the outer 
circle de. A crank is used for converting the forward and 
backward motion of the piston into circular motion; the 
eccentric, on the other hand, is usually employed for convert- 
ing the circular motion of the main shaft back into rectilinear 
motion. With an eccentric, the distance ac, Fig. 121 (B), 
corresponding to the length of the crank arm, may be as small 
as we please. The most frequent uses: to which eccentrics are 
put are to drive slide valves and pumps, the travels of which 
are very much less than that of the piston. 

The distance ac, from the center of the shaft to the center of 
the eccentric, is called the half -throw or the eccentric radius of 
the eccentric, and is equal in length to the half-travel of the 
purl to be driven, such as the pump plunger, or slide-valve. 

Fig. 120 repress nts side elevation and a longitudinal section 
of an eccentric and rod as used fer driving an ordinary slide- 
valve. The circular portion a, which corresponds to the crank 
pin. is called the sheave of the eccentric. It is usually made of 
cast iron in two halves, which are bolted together round the 
shaft, and keyed on in the proper position The piece bb is 
called the strap, and corresponds with the big end of a connect- 



262 Hand Book of Calculations. 



THE ECCENTRIC. 

ing rod. The strap is made of cast iron, or steel, according to 
circumstances, and is lined with a brass or white metal ring, 
where it comes in contact with the sheave. This ring is 
grooved, as shown in the section at cc, so as to prevent it from 
getting off the sheave. The strap is made in two halves bolted 
together so that it can be readily put on, or taken off the sheaf. 





The ring around the eccentric is called the eccentric strap. 

The rod connecting the strap to the part to be put in motion 
is the eccentric rod. 

The hook at the end of the rod, by . which it is connected 
with the rock- shaft of the valve motion is the eccentric hook, or 
gad. 

The whole apparatus is the eccentric-gear. 

Note. 
Eeciprocating motion is motion alternately up and down or 
backwards and forwards like the action of a piston rod. 

Ceaxk Shafts. 

The shaft of the engine is the part which receives circular 
motion from the crank and the reciprocating pieces. By 
means of the shaft, the power generated in the cylinder is 
transmitted to the machinery intended to be driven. Thus, in 
the case of factory engines, a pulley is usually keyed on to the 
shaft, and by means of a leather belt passing over this pulley, 



Hand Book of Calculations. 263 

CRANK SHAFTS. 

the various lines of shafting throughout the building are 
driven. In locomotive engines, the driving wheels are keyed 
direct on to the shaft, and rotate with it, and in the case of 
marine engines, the paddles or screw are also attached directly 
upon the shaft or its prolongation. 

Shafts are subjected to a variety of strains. In the first place, 
they undergo bending stresses from any weights which may be 
attached to them, the most considerable of which is that of the 
fly-wheel, acting vertically downwards. Also the pull of the 
driving belt causes a bending stress, which acts in the line 
joining the driving and the driven shaft. The most important 
stresses, however, arc due to the direct thrust and pull of the 
connecting rod, or rods, which, when at their maximum, act 
m the line of the axes of the cylinders. 

Jourxals. 

The part of the shaft which is supported by the bearing is 
called the journal. The usual form of the journal of an engine 
crank is shown in Fig. 122. The part which runs in the bear- 
ings is turned so as to be truly cylindrical. 
The end play of the shaft is limited by the 
two raised collars. The length of the journal, 
or the distance between the inner faces of the 
collars, relatively to the diameter depends prin- Fl »- 122 - 
eipally upon the number of revolutions which the shaft has to 
make per minute. For slow-running engines the length is 
sometimes equal to the diameter, whereas in cases of high speed 
it may be as much as from two to three times the diameter of 
the journal. 

it care must be taken in designing journals not to pass 
abruptly from one section of the metal to another. All such 
differences should be gradually rounded off as shown in Fig. 122. 

The strains to which the journals of crank shafts are subjected 
lue to the combined action of the twisting forces and the 
transverse loads. 



Xc 



264 



Hand Book of Calculations. 



Shaft Beaki^gs axd Pedestals. 

The bearing usually consists of brass steps supported by a 
cast-iron pedestal or plummer block. Fig. 123 shows three 




Fig. 123. 
views in half elevation and half section of a common form of 
pedestal which is used with a masonry foundation. It consists 
of a wall plate which is bolted to the foundation and on which 
is fixed the pedestal proper. The nature of the arrangement 
and the means by which the steps are adjusted and secured are 
sufficiently explained by the drawing. 

In most stationary engines one or both of the pedestals are 
attached to the cast-iron framework as shown iu Fig. 124, 

which represents the principal 
pedestal of a horizontal engine. 
In this case the steps are not 
divided horizontally, but in an 
oblique plane, so that the direc- 
tion of the resultant force of the 
pull cr thrust in the connecting 
rod and of the other forces which 
act on the shaft, may pass through 
Fig. 124. the solid metal of the step and not 

through the junction between the steps. 

In the case of locomotives the bearings are not fixed, but are 
free to slide up and down in a vertical plane, within the limits 
allowed by the springs. These bearings are- called axle boxes. 
The whole weight of the engine is transmitted through them 
to the journals by means of the springs. 




Hand Book of Calculations. 



265 



Governors. 

If, during the working of a steam engine, the load were 
wholly or partially removed while the supply of steam to the 
cylinder remained undiminished, the engine would commence 
to race. If, on the contrary, the load were increased, the speed 
of the engine would be reduced below the proper rate. To 
prevent such variations in the speed, a contrivance called a 
governor is made use of which acts upon the steam supply in 
one of two ways; viz., either by partially closing or opening the 
throttle valve which regulates the flow of steam from the boiler; 
or else, by acting directly on the valve gear in such a way as to 
vary the point in the stroke where the steam is cut off, and 
thus alter the rate of expansion. 

The most common form of governor was invented by AYatt. 
It consists (see Fig. 125) of two heavy metal balls A, D, 
attached to two inclined arms, which 
latter are jointed at the point E, to 
the central vertical spindle. The 
latter is connected by gearing with 
the main shaft of the engine so as to 
revolve at a rate strictly proportional 
to that of the shaft. The effect of ' 
rotation is that the balls tend to fly 
away from the vertical spindle and, 
being controlled by the arms, they 
can only rise and fall in arcs of cir- === 
cles about the center E. Supposing 
that the velocity of rotation were in- 
creased beyond the normal rate, the 
balls would fly out and occupy some new position D', at the 
same time lifting the collar II which slides on the central 
spindle and which is attached by the links L and K and to the 
ball arms M and \. Into the collar 11 gears the forked end of 
a bell crank lever which is connected by a link with the throttle 
valve. When H is lifted the link acta upon the throttle valve, 
partly closing it. and reducing the supply of steam; on the 
other hand when the balls fall. II falls also and the throttle 
valve is opened. 




266 



Hand Book of Calculations, 



GOVERNORS. 

Many advantages are found to attend the use of high-speed 
governors. They are more sensitive to alterations in speed, the 
parts may be made lighter and move with less friction. In 
order, however, to prevent the balls from flying out too far, in 
consequence of the increased speed of rotation, a weight, or 
else a spring is so arranged as to act on the ball arms in such a 
manner as to develope a radial force in the contrary direction 
to the line of action of the centrifugal force. Fig. 126 shows a 

loaded high-speed gover- 




/l\ 



D 



nor. Each ball is attached 
to two sets of links. The 
weight is arranged to slide 
on the central spindle, and 
t b presses directly upon the 
lower pair of ball links. 

The forms of governors 
are so numerous that it has 
been impossible here to do 
more than explain the prin- 
ciples upon which they act. 



Fig. 126. 



Locomotive engines are 
never fitted with governors, 
but in marine engines they are very necessary, as racing may 
ensue whenever the propeller is partially out of water, or when- 
ever the propeller or crank shaft may give way. On account of 
the motion on board ship, the forms of governors used on land 
engines could not be employed for marine purposes. Marine 
governors are of two principal sorts, viz. those that are actuated 
by variations in the water pressure at the stern of the ship, and 
those which depend for their motion on variations in the velocity 
of the engine. The former class only provide for cases due to 
the incomplete immersion of the propeller, but the latter will 
guard against every contingency. In consequence of the great 
size of the throttle valves and expansion gear of marine engines, 
an ordinary governor cannot conveniently be employed to act 
directly on the controlling parts; hence, in this class of engines, 
what are called steam governors are now generally employed. 



Hand Book of Calculations, 



267 



GOVERNORS. 

The governor proper is arranged to move the slide valve of a 
small steam cylinder, which, in its turn, actuates the throttle 
valve. 



Fly-AViieels. 




The fly-wheel is a wheel hav- 
ing a heavy run. It is gener- 
ally keyed to the crank axle of 
the engine, and is used for mod- 
ifying the effects of any irregu- 
larity either in the driving 
power or in the resistance to be 
overcome. When, for instance, 
the driving power is in excess 
of the resistance to be overcome, 
the surplus is expended in in- 
Fl £* **'• creasing the velocity of the fly- 

wheel; and, vice versa, when the resistance is in excess of the 
drivingpower, the energy stored up in the fly-wheel is expended 
in helping to overcome the resistance, during which operation 
its velocity is lowered. 



The greater portion of the mass of a fly-wheel is concentrated 
in its rim, and when revolving, every particle of the rim is 
under the action of centrifugal force, and tends to fly away 
radially from the center; hence the rim, when in a state of 
revolution, resembles the condition of a ring put in a state of 
tension by a force from within acting outwards. The tension 
developed in the rim is opposed by the tensile strength of the 
metal of which it is formed, and should the former exceed the 
latter the rim will inevitably burst asunder, just as a boiler 
would burst if the steam pressure were too great for the strength 
<»f the shell plates. 



268 



Hand Book of Calculations, 



ENGINE COUNTERS. 



The following representation (Fig. 128) shows one form of an 
engine counter. It, or some other form of a counter, is used 
upon nearly all marine engines and very many land engines. 




Fig. 128. 

The operation of the device is simple and accurate. Each 
revolution of the engine moves the cogs of the right hand dial 
one notch, and upon the 10th stroke of the engine, it moves the 
pointer on the second dial to the figure one — all the dials at the 
beginning being placed at the zero. "When the pointer on the 
second dial has completed nine and passes to the zero then at 
the same instant, the third dial registers one, indicating that 
the engine has completed 1,000 revolutions. 

Whatever number of dials (or wheels) there may be, the right 
hand always records 10, the next to the left 100, the 3rd, 1,000; 
the 4th, 10,000; the 5th, 100,000; and the 6th, 1,000,000. 
Hence, the above having 6 dials, can register 1,000,000 revolu- 
tions. 

What the pointers point at in the above, beginning at the 
left, and reading towards the right; 5, 0, 6, 0, 4, and 9 is 
506049 revolutions. 

Fig. 129 is the form now generally used; it has 7 dials, and 
is read from left to right, 9879460. From zero to zero will be 
10,000,000. 




asBassco 



u 



Fig. 129. 



Hand Book of Calculations. 269 

ENGINE COUNTERS. 

When a counter has completed its full number of recording, 
all the numbers will show zero, to which must be added an 
imaginary one, making for 7 dials or wheels, ten millions — and 
the next stroke of the engine will begin a new series with 1, 

etc 

Example of use of the Exgixe Couxters. 
An engine counter stood at 90 7 at the commencement of a 
voyage, and after 9 days, 3 3 hours, 15 minutes, 42 seconds 
stood at 590049: how many revolutions per minute have been 
made by the engines ? 

d. h. m. s. 

From 5960-49 Reduce 9 13 15 42 to minutes. 



Take 90T 24 



595082 229 decimal of a minute: Thus, 
60 If =.7; and place this .7 

13755.7 



Bring the 42 sees, to the 
decimal of aminul 
If =.7; and place 
after the minutes, 



Then, 595082^13755.7=43.26. 

Answer, 43.26 revolutions per minute. 



ILLUMINATING GAS. 



Tlie unit for measurement of light, either electric, gas, oil, or 
tallow light, is called one candle power. Light can be meas- 
ured with great accuracy, owing to an invariable law, which is 
similar to the law of gravitation. 

If two lights of unequal power be made to shine on the sur- 
face of a smooth plaster wall, and a book or card be interposed, 
the two shadows produced by the crossing of the rays will differ 
in blackness in the same degree as the powers of the two lights; 
the stronger light will produce the darker shadow. 

To obtain the difference in power of the two lights, the 
stronger light mnsl be moved backwards or the lesser light for- 
ward until both shadows are the same tint, wdiich the eye can 
tell to great exactness. 



2jo Hand Book of Calculations. 

ILLUMINATING GAS. 

The Eule of Light. 
The intensity, and consequent value of light is as the square 
of its distance. 

Example. 
Suppose a candle be 6 feet from the wall and a gas light 12 
feet, to be of equal shades, then 6 1 = 36 and 12 1= =144 making 
the gas light 4 times as great as the light of the candle, or one 
gas light to be equal to the light of four candles. 

Example. 
What amount of light at 10 feet distance will equal that of 
one candle at 2 feet distance from the point of equal light ? 
Now, then: The square of 2 (2x2) is 4, and the square of 10 
(10x10) is 100; so the amount of light is as 100 is to 4, or 
(100-^4) equals 25 — the number of candles required at a dis- 
tance of 10 feet to produce a light equal to one at 2 feet. 

The Photometee and the Unit of Light. 
The illuminating power of gas, electricity, oil, etc., is measured 
by an instrument called the photometer, and the unit of meas- 
urement as fixed by law and custom is the consuming of 120 
grains per hour of a sperm candle, of ivhich it takes six to make 
one pound. Deficiency of light and all impurities are shown 
by the instrument when reduced to the test of comparison here 
described. 

Table showing the amount of Oxygen consumed by 
equal Lights. 

Tallow candles consume 12.0 feet of oxygen gas. 

Wax " " 8.4 " " 

Paraffin oil " .....6.8 " 

Coal-gas " 5.4 

Carbonized gas " 3.3 

Table of the Cost of Equal Light feom Diffeeent 

Mateeials. 
21 ft. of gas costing 2 cts. =1 lb. of dip candles costing 12c. 
25 " " " 2| " =1 " of composite candles " 16c. 
25 *'< " " 2| " =1 " of wax candles u 41c. 

175" " " 18 " =1 gallon of paraffin " 100. 






Hand Book of Calculations. 



211 



ILLUMINATING GAS. 

Table of Heat produced by different Lights of equal 

Power. 

Tallow candles 505 lbs. of water warmed 10° 

Wax " 383 '• 

Paraffin oil 3«1" " 

Coal gas 278" " " " 

Carbonized gas 195 " " " " 

The Gas Meter. 

The unit of quantity, for gas, is the cubic foot, and the machine 
for accomplishing the measurement is called the Gas Meter, of 
which there are two kinds, the wet meter, and the dry meter. 
Afl both are measures of volume, there is no difference between 
them, so far as economy is concerned, any more than when the 
measurements of liquids are effected by a copper or tin vessel. 

The Index of the Gas Meter. 
Fig. 130 is a drawing of the ordinary index used for meters 




Fig. 130. 

supplying up to 10 lights. When a larger capacity than these 
the index is provided with 4 and sometimes 5 dials. The 
hand on each index as shown in the figure moves round in same 
direction as the figures count; thus from 1 to 2 to 3, etc. The 
hand on the index (beginning at the left) moves to the right, 
that of the second turns to the left, that of the third turns to 
the right. In a new meter all the hands point to the cypher 
(0) at the top, which shows that no gas has been used. When 



2J2 Hand Book of Calculations. 

HOW TO ftEAD A GAS METEE. 

the hand on the right index has moved to 1 it implies that 100 
feet of gas have been used; when it reaches 6 it means 600 feet; 
and on completing the circuit at the top (0) it is 1000 feet: 
each time this hand passes round 1000 feet of gas have been 
used. Each of the other indexes are tenfold multipliers of each 
other. Single figures are placed on the face for want of room; 
thus, in the first 1 means 100; in the second 1 stands for 1000; 
in the third 1 means 10,000; and so on with the succeeding 
figures respectively. "Cents" is sometimes placed over the 
index to imply that 100 feet is the smallest quantity shown. 

To read the meter, begin with the left index and write down 
the quantity last past ty the hand, then write down the quan- 
tity last past on the second index; and proceed with the third 
in the same manner, when the figures as shown in the above 
illustration will be 28,600, being the cubic feet of gas used at 
that time; read the meter again next day (or at any other time) 
and the figures will read— say 29,300 feet; deduct the previous 
quantity from this amount, and the difference, 700 feet, is the 
quantity used since the last reading. If the hand of any index 
appears to be exactly on any one figure, you must refer to the 
hand of the next index to the right of it, and this will show if 
the other hand has passed the figure in doubt or not. Thus, if 
the second hand has completed its revolution, it implies the 
first has passed the figure; if the second has not completed its 
revolution, it implies the first has not passed the figure. This 
must be carefully observed to prevent mistakes. 

There is sometimes a small index above the others, showing 
single feet. Generally 5 feet are indicated by each revolution, 
as in the above illustration; but no account of this is kept by 
the gas companies; it is only for experimental purposes; for 
instance, if you wish to know how much is used per hour or 
minute. Ten minutes" practice at reading meters (called taking 
the meter) will render the matter quite familiar, and it will be 
well for the engineer in charge to keep a daily register of the 
meter, as is done in most large establishments; it not only de- 
tects any wasteful use of gas by escape or otherwise, but will 
also show if the meter continues to work properly. 



Hand Book of Calculations. 273 

Points Relating to Gas Important for the Engineer 

to Know. 

An engineer should have the same supervision of the gas 
consumed on the premises, as he does of the oil that is used, 
the fuel burned, etc., as a matter both of economy and of 

safety. 

The lights in large establishments are generally turned out at 
the main and the burners turned off afterward. A saving of 
_ a is effected by this system from two reasons, namely, by 
putting it out more quickly, and by preventing any escape 
from the fittings when not in use. Others prefer turning off 
the burners only, so as to have the gas always ready to light, if 
wanted in the night. As numberless explosions have happened 
through turning off at the main, it is safest to keep the gas 
altcays on. 

The effect of distance being so important in matters of light 

four times the amount of light is required by only doubling 

the distance at which it is placed), lights should therefore be 

placed as near the object to be seen as conveniently may be, 

and not shine in the eyes of the observer. 

It must be borne in mind that the meter indicates the quan- 
titv of gas which passes without any reference as to how it is 
. and care must be used, in preventing its waste and loss. 

All the pipes placed in inaccessable places, such as beneath 
the flooring, should be of wrought iron to prevent " bagging" 
or falling into recesses, at intervals, in which the vapors often 
condense and obstruct the passage of the gas. 

Every size of burner requires a definite amount of gas to pro- 
duce the largest proportion of light: light is as much sacrificed 
by using too little gas, as by using too much. 

No burner ought to be used out of doors without being pro- 
-1 by glass; sufficient air must nevertheless be admitted to 
all burners, or the combustion will be imperfect, the color of 
the light will be bad, and smoke will be produced. 

Flickering is principally caused by insufficient pressure. 



2J4 Hand Book of Calculations. 

ILLUMINATING GAS. 

A greater amount of light is produced from one large burner 
than from two small ones consuming the same quantity as the 
one large one. An argand burner consuming five feet of gas 
per hour under one-tenth of an inch pressure will produce the 
light of twelve candles; but a similar burner, so small as to 
require a pressure of one inch to consume the same quantity of 
gas, will only produce one-fourth that amount of light. 

Whenever there is an odor of escaping gas emanating from 
the street, cellar, drain, cistern, sewer, or anywhere in immed- 
iate neighborhood of the consumer's premises, written notices 
should be sent without delay to the company, who will immed- 
iately attend to it for their own protection. Should there be 
signs of an escape in the interior of a building, immediate and 
prompt care must be employed. Lights of any hind should be 
avoided, the main tap turned off, the doors and upper parts of 
the windows opened (as gas by its lightness ascends and escapes 
very readily at the highest part of an apartment.) When the 
source of escape has been found, it can be temporarily stopped 
with a little grease, white lead or soap and afterwards substan- 
tially repaired. A room can be more safely entered by crawling 
upon the hands and knees than by walking upright. 
The Engineer's Signal Code. 

The sign O means a short, quick sound, while the dash — 

means a long sound. 

Apply brakes, stop O 

Eelease brakes, start O O 

Back OOO 

Highway crossing signal OO or OO 

Approaching station, — blast lasting 5 seconds. 

Call for switchman OOOO 

Cattle on track 

Train has parted — O 

Railroad crossing, same as approaching station. 

For fuel OOOOO 

Bridge or tunnel warning OO — 

Fire alarm ... — OOOO 

Will take side track 

Bed signifies danger; green signifies caution, go slowly; 

green and white signifies stop at flag stations for orders, for 

passengers or freight. One cap or torpedo on rail means stop 

immediately; two caps or torpedoes means reduce speed im- 

. mediately and look out for danger signal. 



Hand Book of Calculations 



215 



TABLE 

Of Squares, Cubes, Square and Cube Roots of Numbers. 



Number. 


Square. • 


Cube. 


Square Root, 


Cube Root. 


1 


1 


1 


1.0 


1.0 


a 


4 


8 


1.414213 


1.25992 


3 


9 


27 


1.732050 


1.44225 


4 


16 


64 


2.0 


1.58740 


.-) 


26 


125 


2.236068 


1.70997 


6 


36 


216 


2.449489 


1.81712 


i 


49 


343 


2.645751 


1.91293 


8 


64 


512 


2 828427 


2.0 


9 


81 


729 


3.0 


2.08008 


10 


100 


1000 


3.162277 


2.15443 


11 


121 


1331 


3.316624 


2.22398 


12 


144 


1728 


3.464101 


2.28942 


13 


169 


2197 


3.605551 


2.35133 


14 


196 


2744 


3.741657 


2.41014 


15 


223 


3375 


3.872983 


2.46621 


16 


256 


4096 


4.0 


2.51984 


17 


289 


4913 


4.123105 


2.57128 


18 


324 


5832 


4.242640 


2.62074 


19 


361 


6859 


4.358898 


2.66840 


20 


400 


8000 


4.472136 


2.71441 


21 


441 


9261 


4.582575 


2.75892 


22 


484 


10648 


4.690415 


2.80203 


23 


529 


12167 


4.795831 


2.84386 


24 


576 


13824 


4.898979 


2.88449 


26 


025 


15625 


5.0 


2.92401 


26 


676 


17576 


5.099019 


2.96249 




729 


19683 


5.196152 


3.0 


28 


784 


21952 . 


5.291502 


3.03658 


29 


841 


24389 


5.385164 


3. 07231 


30 


900 


27000 


5.477225 


3.10723 


81 


961 


29791 


5.567764 


3.14138 




1024 


32708 


5.656854 


3.17480 


88 


1069 


85987 


5.744563 


3.20753 




1156 


39304 


5.880951 


8.23961 




1225 


42875 


5.916079 


3.27106 




1296 


46650 


6.0 


3.30192 




1369 


50658 


6.082762 


3.33222 


N 


1444 


5 4872 


6.164414 


3.36197 




1521 


59319 


6 244998 


3.39121 


40 


1600 


64000 


6.324555 


3.41995 



2j6 



Hand Book of Calculations. 



TABLE— {Contiuued) 

OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS. 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


41 


1681 


68921 


6.403124 


3.44821 


42 


1764 


74088 


6.480740 


3.47602 


43 


1849 


79507 


6.557438 


3.50339 


44 


1936 


85184 


6.633249 


3.53034 


45 


2025 


91125 


6.708203 


3.55689 


46 


2116 


97336 


6.782330 


3.58304 


47 


2209 


103823 


6.855654 


3.60882 


48 


2304 


110592 


6.928308 


3.63424 


49 


2401 


117649 


7.0 


3.65930 


50 


2500 


125000 


7.071067 


3.68403 


51 


2601 


132651 


7.141428 


3.70843 


52 


2704 


140608 


7.211102 


3.73251 


53 


2809 


148877 


7.280109 


3.75628 


54 


2916 


157464 


7.348469 


3.77976 


55 


3025 


166375 


7.416198 


3.80295 


56 


3136 


175616 


7.483314 


3.82586 


57 


3249 


185193 


7.549834 


3.84850 


58 


3364 


195112 


7.615773 


3.87087 


59 


3481 


205379 


7.681145 


3.89299 


60 


3600 


216000 


7.745966 


3.91486 


61 


3721 


226981 


7.810249 


3.93649 


62 


3844 


238328 


7.874007 


3.95789 


63 


3969 


250047 


7.937253 


3.97905 


64 


4096 


262144 


8.0 


4.0 


65 


4225 


274625 


8.062257 


4.02072 


66 


4356 


287496 


8.124038 


4.04124 


67 


4489 


300763 


8.185352 


4.06154 


68 


4624 


314432 


8.246211 


4.08165 


69 


4761 


328509 


8.306623 


4.10156 


70 


4900 


343000 


8.366600 


4.12128 


71 


5041 


357911 


8.426149 


4.14081 


72 


5184 


373248 


8.485281 


4.16016 


73 


5329 


389017 


8.544003 


4.17933 


74 


5476 


405224 


8,602325 


4.19833 


75 


5625 


421875 


8.660254 


4.21716 


76 


5776 


438976 


8.717797 


4.23582 


77 


5929 


456533 


8.774964 


4.25432 


78 


6084 


474552 


8.831760 


4.27265 


79 


6241 


493039 


8.888194 


4.29084 


80 


6400 


512000 


8.944271 


4.30887 1 



Hand Book of Calculations. 



2 77 



TABLE— (Continued) 

OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS. 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


81 


6561 


531441 


9.0 


4.32674 


82 


6724 


551368 


9.055385 


4.34448 


83 


6889 


571787 


9.110438 


4.36207 


84 


7056 


592704 


9.165151 


4.37951 


85 


7225 


614125 


9.219544 


4.39683 


86 


7396 


636056 


9.273618 


4.41400 


87 


7569 


658503 


9.327379 


4.43104 


88 


7744 


681472 


9.380831 


4.44796 


89 


7921 


704969 


9.433981 


4.46474 


90 


8100 


729000 


9.4868S3 


4.48140 


91 


8281 


753571 


9.539392 


4.49794 


92 


8464 


778688 


9.591663 


4.51435 


93 


8649 


804357 


9.643050 


4.53065 


94 


8836 


830584 


9.095359 


4.54683 


95 


9025 


857375 


9.746794 


4.56290 


9G 


9216 


884736 


9.797959 


4.57785 


97 


9409 


912673 


9.848857 


4.59470 


98 


9604 


941192 


9.899194 


4.61043 


99 


9801 


970299 


9.949874 


4.62606 


100 


10000 


1000000 


10.0 


4.64158 


101 


10201 


1030301 


10.049875 


4.65701 


102 


10404 


1061208 


10.099504 


4.67233 


103 


10609 


1092727 


10.148891 


4.68754 


104 


10816 


1124864 


10.198039 


4.70266 


106 


11025 


1157625 


10 246950 


4.71769 


100 


11236 


1191016 


10.295630 


4.73262 


107 


11449 


1225043 


10.344080 


4.74745 


108 


11664 


1259712 


10.392304 


4.76220 


109 


11881 


1295029 


10.440306 


4.77685 


110 


12100 


1331000 


10.488088 


4.79142 


111 


12321 


1367631 


10.535653 


4.80589 


112 


12544 


1404928 


10.583005 


4.82028 


113 


12769 


1443897 


10.630145 


4.83458 


114 


12996 


1481541 


10.077078 


4.84880 


115 


13225 


1520875 


10.723805 


4.86294 


116 


13456 


1500896 


10.770329 


4.87699 


117 


13689 


1601613 


10.810653 


4.89097 


118 


1 8924 


16430:52 


10.862780 


4.94086 


119 


14161 


1685159 


10.008712 


4.91868 


120 


14400 


1728000 


10.954451 


4.93242 



278 



Hand Book of Calculations. 



TA BLJE— (Continued) 

OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS. 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


121 


14641 


1771561 


11.0 


4.94608 


122 


14884 


1815848 


11.045361 


4.95967 


123 


15129 


1860867 


11.090536 


4.97319 


124 


15376 


19066-24 


11.135528 


4.98663 


125 


15625 


1S53125 


11.180339 


5.0 


126 


15876 


2000376 


11.224972 


5.01329 


127 


16129 


2048383 


11.269427 


5.02652 


128 


16384 


2097152 


11.313708 


5.03968 


129 


16641 


2146689 


11.357816 


5.05277 


130 


16900 


2197000 


11.401754 


5.06579 


131 


17161 


2248091 


11.445523 


5.07875 


132 


17424 


2299968 


11.489125 


5.09164 


133 


17689 


2352637 


11.532562 


5.10446 


134 


17956 


2406104 


11.575836 


5.11723 


135 


18225 


2460375 


11.618950 


5.12992 


136 


18496 


2515456 


11.661903 


5.14256 


137 


18769 


2571353 


11.704699 


5.15513 


188 


19044 


2628072 


11.747344 


5.16764 


139 


19321? 


2685619 


11.789826 


5.18010 


140 


19600 


2744000 


11.832159 


5.19249 


141 


19881 


2803221 


11.874342 


5.20482 


142 


20164 


2863288 


11.916375 


5.21710 


143 


20449 


2924207 


11.958260 


5.22932 


144 


20736 


2985984 


12.0 


5.24148 


145 


21025 


3048625 


12.041594 


5.25358 


146 


21316 


3112138 


12.083046 


5.26563 


147 


21609 


3176523 


12.123455 


5.27763 


148 


21904 


3241792 


12.165525 


5.28957 


149 


22201 


3307949 


12.266555 


5.30145 


150 


22500 


3375000 


12.247448 


5.31329 


151 


22801 


3442951 


12.288205 


5.32507 


152 


23104 


3511808 


12.328828 


5.33680 


153 


23409 


3581577 


12.369316 


5.34848 


154 


23716 


3652264 


12.409673 


5.36010 


155 


24025 


3723875 


12.449899 


5.37168 


156 


24336 


3796416 


12.489996 


5.38323 


157 


24649 


3869893 


12.529964 


5.39469 


158 


24964 


3944312 


12.569805 


5.40612 


159 


25281 


4019679 


12.609520 


5.41750 


160 


25600 


4096000 i 


12.649110 


5.42883 



I land Book of Calculations. 



19 



TABLE— (Continued) 

OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS. 



161 
162 

163 

164 
16~) 

166 
167 
168 
169 
170 

171 
172 
173 
174 

17o 

176 
177 
178 
179 
180 

181 
182 
183 
184 
185 

186 

188 

190 

191 
192 
198 

194 
196 

196 

197 
198 

199 



Square. 


Cube. 


Square Root. 


Cube Root. 

5.44012 


25921 


4173281 


12.688577 


26344 


4251528 


12.727922 


5.45136 


26569 


4330747 


12.767145 


5.43255 


26896 


4410944 


12.806248 


5.47370 


27225 


4492125 


12.845232 


5.48480 


27556 


4574296 


12.884098 


5.49586 


27889 


4657463 


12.922848 


5.50687 


28224 


4741632 


12.961481 


5.51784 


28561 


4826809 


13.0 


5.52877 


28900 


4913000 


13.038404 


5.53965 


29241 


5000211 


13.076696 


5.55049 


29584 


5088448 


13.114877 


5.56129 


29929 


5177717 


13.152946 


5.57205 


30276 


5268024 


13.190906 


5.58277 


30625 


5359375 


12.228756 


5.59344 


30976 


5451776 


13.266499 


5.60407 


31329 


5545233 


13.304134 


5.61467 


31684 


5639752 


13.341664 


5.62522 


32041 


5735339 


13.379088 


5.63574 


32400 


5832000 


13.416407 


5.64621 


32761 


5929741 


13.453624 


5.65665 


33124 


6028568 


13.490737 


5.66705 


33489 


6128487 


13.527749 


5.67741 


33866 


6229504 


13.564660 


5.68773 


34225 


6331625 


13.601470 


5.69801 


34596 


6434856 


13.638181 


5.70826 


34969 


6539203 


13.674794 


5.71847 


3634 1 


6644672 


13.711309 


5.72865 


35721 


675126!) 


13.747727 


5.73879 


96100 


6859000 


13.784048 


5.74889 


36481 


6967871 


13 820-3; 5 


5.75896 


8686 1 


7077888 


13.856406 


5.76899 


37249 


71X9057 


13.892444 


5.778S.9 


37686 


7301384 


18.928388 


5.78896 


88026 


7114875 


13.964240 


5.79889 


B8416 


7529536 


14.0 


5.80878 




7615373 


14.035668 


5.81864 


39204 


7762892 


11.071247 


5.82847 


89601 


7880599 


14.106786 


5.88827 


40000 


8000000 


14.142185 


5.84808 



280 



Hand Book of Calculations. 



MELTING POINTS OF SOLIDS. 



The metals are solid at ordinary temperatures, with the ex- 
ception of mercury, which is liquid down to — 39° F. Hydro- 
gen, it is believed, is a metal in a gaseous form. 

All the metals are liquid, at temperatures more or less eleva- 
ted, and they probably turn into gas or vapor at very high 
temperatures. Their melting points range from 39 degrees 
below zero of Fahrenheit's scale, the melting, or rather the 
freezing, point of mercury, up to more than 3000 degrees, 
beyond the limits of measurement by any known pyrometer. 
Certain of the metals, as iron and platinum, become pasty and 
adhesive at temperatures much below their melting points. 
Two pieces of iron raised to a welding heat, are softened, and 
readily unite under the hammer; and pieces of platinum unite 
at a white heat. 

Melting Points of Solids. 



VARIOUS SUBSTANCES. 



Sulphurous acid 
Carbonic acid. . 

Bromine , 

Turpentine 
Hyponitric acid , 

Ice , 

Nitro-glycerine. 

Tallow 

Phosphorus 

Acetic acid 

Stearine 

Margaric acid. . , 
Wax, rough. . . . 
" bleached 

Iodine 

Sulphur 



Melting Points. 



-148° F. 
-108 
+9. 5 
14 
16 
32 
45 
92 
112 
113 
109 to 120 
131 to 140 
142 
154 
225 
239 



Hand Book of Calculations. 



281 



Melting Points of Solids. — {Continued.) 


METALS. 


Melting Points. 


Mercury 


-39° F 
144 
208 
356 
442 
507 
617 
680 to 773 
810 to 1150 

1692 

1832 to 1873 

1996 

2156 

1922 to 2012 

2012 to 2786 

2372 to 2552 

2732 

2912 


Potassium 


Sodium 


Lithium 


Tin 


Bismuth 


Lead 


Zinc 


Antimony. . 


Bronze 


Silyer 


Copper. . „ 


Gold, standard 


Cast Iron, white 


" li gray 


Steel 


A\ rought Iron. . , , 


Hammered Iron 





SUNDRY ALLOYS OF TIN, LEAD, AND BISMUTH. 


Melting Points. 


3 Lead, 2 Tin, 5 Bismuth 


199° 

201 
212 
246 
334 
334 
360 to 385 
392 
552 


1 " 1 " 4 " 


5 " 3 " 8 " 


1 " 4 " 5 " 


1 " 3 


•> " 1 " 


1 " 2 " 


3 " 1 « 


3 " 1 " 



ALLOTS FOR FFSIBLE PLUGS. 


Softens at 


Melts at 


•1 'I'm. 2 Lead 


365° F. 

372 

395J 


372° F. 
383 
406 to 410 


2 " 6 " 

2 " 8 " 





282 



Hand Book of Calculations. 



THE BAROMETER 

Consists of a glass tube closed at one end, a cup, and some 
mercury. 

It will be noticed that the mercury in the cup is exposed to 
the pressure of the atmosphere, whereas that in the tube is not. 

In considering the operation of a low pressure engine suppose 
the vacuum in the condenser to be perfect, the atmosphere 
pressing on the mercury will force it up the leg of the tube till 
it stands at 30 inches; therefore 30 inches of mercury means a 
perfect vacuum. As vapor increases in the condenser, it will 
flow on the top of the mercury and force it down. 

Again, when there is a perfect vacuum in 
the condenser or under the piston, if the at- 
mosphere could be allowed to act on the upper 
face of the piston, it would force it down 
with a pressure of 15 lbs. per square inch; 
and as vapor arises it flows up the eduction 
pipe under the piston, and destroys so much 
of the atmospheric pressure. Hence, if the 
Barometer stands at 30 inches, we speak cf 15 
lbs. of vacuum; if at 28 inches, 14 lbs. of 
vacuum; and if at 25 inches, 12-g- lbs. of 
vacuum, and so on. 

Instead of the atmosphere, steam of a press- 
ure greater than that of the atmosphere is 
admitted to the piston; thus, if steam of 25 
lbs. pressure is admitted, we speak of it as 10 
lbs. above the atmosphere. If there were a 
Fig. 131. perfect vacuum, it would exert 25 lbs. press- 

ure on the piston; but if the mercury be at 2 G inches, a press- 
ure equal to 4 inches or 2 lbs. will be lost, or the effective 
pressure will be 23 lbs. per square inch, or as it is usually 
expressed: 

10 lbs. of steam. 
26 inches =13 lbs. of vacuum. 

23 lbs. effective pressure. 
A vacuum gauge is now used instead of a Barometer. 




Hand Book of Calculations. 283 

THE BAROMETER. 

The weather Barometer is the instrument for showing the 
pressure of the air; when it stands at 30 inches, if pure water 
be boiled in the open air, and the Thermometer placed in it 
while boiling, the mercury will invariably stand at 212°. Sea 
water boiled under the same circumstances will send the mer- 
cury up to 213.2, or 1.2° more than pure water. This 1.2° 
must be due to the salt contained in the sea water, hence, for 
every degree of salt we may reckon an increase of 1.2° in the 
temperature of the boiling point, the Barometer remaining at 
30 inches. 

The following table shows the correct boiling point of salt 
water at the different degrees of density, when the Barometer 
stands at 30 inches. 



Boilixg Poixt of Salt Watek. 

Degrees 

Fresh Water AV boiling point 212 

Sea Water & solid matter 

A " 

" 

A " 



rs 

_6 
I :; 

7 
":; I 

s 

n 

B 

~.i f 

i 

:; I 



Saturated. Salt / 1? (i 

deposited f ** 

Water that has A part salt, has 5 oz. of salt per gall, of water. 
A " 10 oz. 

" " A " 15 oz. " " 

In this table the Imperial Gallon is used. 



(< 


213.2 


(( 


214.4 


il 


215.5 


a 


216.7 


it 


217.9 


a 


219.1 


a 


220.3 


iC 


221.5 


te 


222.7 


a 


223.8 


a 


225.0 


a 


226.1 


1. of water. 




(S 




(( 



284 Hand Book of Calculations. 



THE HYDEOMETEE. 

Hydrometer is the general name given to instruments for 
testing the density of liquids; but when the instrument is spec- 
ially marked for testing the density of some particular liquid, 
it has a special name given to it; thus, the Hydrometer spec- 
ially marked for testing the amount of water in milk, is called 
a Lactometer, meaning a "milk-measurer." In the same way 
the Hydrometer specially marked for testing the amount of salt 
in the boiler water, is called a Salinometer, meaning a "salt- 



THE SALINOMETEE. 

A salinometer is a glass or metal instrument by 
means of which the density of water is ascertained. 
It consists of a weighted bulb, to which is attached 
a graduated stem, and its action is to indicate the 
amount of salt held in solution in the water, by 
floating higher or lower; higher for density, lower 
for freshness. Some are graduated into 33rds, and 
some to 32nds, each representing about five ounces 
of salt to a gallon of water. Care must be taken 
to use the salinometer at the temperature for which 
it is marked, as the densities of fluids vary in pro- 
portion to their temperature.' 

200° being the usual temperature of the water in 
which these instruments are tested, so that they 
may be used almost immediately on the water from 
the boiler ceasing to boil. 

Sea-water contains it part salt, that is, if 33 
pounds of sea water were evaporated, 1 pound of 
^J salt would remain; ts is for this reason -taken as 

Fig. 132. the unit by which to measure the density of the 
boiler water. If the water in the boiler has the same amount 
of salt in it as sea- water, we say it has 1 degree of salt; if it 
contains twice as much salt per gallon as sea- water, then it has 
two degrees of salt, that is -iz, &c. 



Hand Book of Calculations. 285 



ELECTRICITY. 



The history of electricity, as a modern science, began nearly 
a centurv ago, when Sir Humphrey Davy, in 1810, exhibited an 
electric light, produced by two small pieces of carbon and a 
powerful galvanic battery, and when, a few years later, Michael 
Faraday arrived at one of the greatest of discoveries, one that 
has been placed on a level with the discovery of the laws of 
gravitation. In brief, the discovery that electricity could 
be produced from magnetism by pow 7 er. Faraday exper- 
imented with a great magnet belonging to the Royal Society 
consisting of 450 bars, each fifteen inches long, and succeeded 
in obtaining a current of sufficient strength to give sparks from 
the end of the wire forming the coils. 

A host of inventors have labored with the principles discov- 
ered by Faraday in endeavoring to reduce them to practical use; 
in 1867 these experiments had so far advanced that the problem 
of electric lighting by means of the dynamo had been assured 
upon a commercial basis, and from that date the progress in 
what may be denominated industrial electricity has been mar- 
vellously rapid. 

Two essential principles may be stated relating to this sub- 
ject: 

1. Electricity as an industrial agent has come to stay; 

2. That the end reached after a century of scientific research 
has resulted in the acknowledgment that electricity is an un- 
known thing; it may be matter, it may be force, or both; but 
however produced, it may be considered one and the same thing. 
Hence, the engineer, who knows he is ignorant upon this mys- 
terious subject is almost on a level with the greatest electrical 
expert. But it may be recalled that until the days of Watt, 
steam was an equally unknown agent, of which even now there 
are some things to be discovered — such as the true nature of 
latent heat, etc., and that some day an instrument like the 
steam engine indicator may be discovered which will illuminate 
the unknown nature of this mysterious agent. 



286 Hand Book of Calculations. 

ELECTRICITY. 

The most accurate definition of electricity perhaps, is that it 
is simply mechanical energy changed into electrical energy, 
that is, that the power existing in the coal as it is consumed 
under the steam boiler, and the zinc, lead, etc., which is con- 
sumed in a primary battery is changed into another form of 
energy known as electricity. 

Since every manifestation of energy is by means of motion, the 
conclusion seems inevitable that electricity is a mode of mole- 
cular motion, and in electricity this motion of the infinitely 
small particles are inconceivably swift, and beyond any piinted 
or verbal description. 

This small sum of accurate knowledge being the result of the 
most advanced science, it follows that the practical engineer 
will wisely confine himself to the duty of reducing to the 
best performance the various mechanisms belonging to his elec- 
tric lighting and power plant. 



Electrical Data and Definitions. 

The student who is determined to add a knowledge of 
industrial electricity to his accomplishments must first learn 
the names, and uses of .electrical appliances, and the definition 
and meaning of electrical terms, and by persistent endeavor 
master the subject from its beginning to its latest develop- 
ment. 

The Dynamo, is the machine in ordinary use for producing 
electricity; when it is thus employed it is called a dynamo- 
electric-machine, but when the same apparatus is used to change 
the electricity into mechanical power it is called a motor. 

The dynamo consists of the following principal parts : 1. The 
armature, which is the revolving portion. 2. The field mag- 
nets, which produce the field within which the armature 
revolves. 3. The 'pole pieces. 4. The commutator ox collector. 
5. The collecting brushes. 



Hand Book of Calculations. 28 j 

ELECTRICAL DATA AND DEFINITIONS. 

There are scores of forms in which dynamos are constructed 
and they are made of varying sizes but all upon the same gen- 
eral principle. 

The dynamo,, like all our electrical machines or batteries, are 
merely instruments for moving electricity from one place to 
another, or for causing electricity when accumulated or 
" bunched up" to do work in returning to its former level dis- 
tribution. 

Electro- mot ire force, sometimes written briefly (E. M. F.) is 
the name which is used to express the force which tends to 
move the electricity from one place to another. 

When electricity is regarded as a fluid, its supposed flow or 
passage is called a current, and any substance through which 
it flows a conductor. Bodies offering such great resistance as 
practically to prevent the passage of electricity are named 
insulators. The path through which a current passes is termed 
a circuit, which when continuous is called a closed circuit, but 
when there is a break in it an open circuit. 

Potential, is a term employed to express various degrees of 
electrical energy, or power of doing work, and is used with 
respect to electricity in the same way as the word pressure is 
applied to steam. A difference of potential (or pressure) 
between two points connected by a conductor, produces a pass- 
age of electricity, which is evidence that the potential of each 
point in the circuit is less than that of each preceding point; 
when there is no difference there ceases to be any transference 
between them. Hence the flow of electricity is like the flow of 
water collected in a reservoir. When there is no " head " there 
is no flowing outward of the fluid. 

In comparing hydraulics and electricity, it must be borne in 
mind that water in pipes has mass and weight, while electricity 
has none — the head or pressure of a stand pipe is what causes 
water to flow through pipes which offer resistance to the flow. 
We might call this pressure water-motive force, so in electricity 
the head or pressure, or as it is called, the electro-motive force 
(E. M. F.) will make the electricity move through the wire. 



288 



Hand Book of Calculations. 



CONDUCTORS AND INSULATORS 
ELECTRICITY. 



OF 



There is no substance so good a conductor of electricity as to 
be devoid of resistance, and there is no substance of so high a 
resistance as to be strictly a non-conductor. 

Hence in the following list the substances named are placed 
in order, each conducting better than those below it on the list. 



Silver. 

Copper. 

Gold. 

Zinc. 

Platinum. 

Iron. 

Tin. 

Lead. 

Mercury. 

Charcoal. 

Acids. 

Water. 



► G-ood Conductors* 



The body. 
Cotton. 
Dry wood. 
Marble. 
Paper. 

Oils. 

Porcelain. 

Wool. 

Silk. 

Resin. 

Gutta Percha. 

Shellac. 

Ebonite. 

Paraffine. 

Class. 

Dry air. 

Worst conductor. 



\ Partial Conductors. 



> Non- Conductors or Insulators. 



Hand Hook of Calculations. 289 



ELECTRICAL DATA AND DEFINITIONS. 

Positive and negative electricity are two convenient terms 
used to express different states in the formation of electricity. 
Bodies charged with electricity of the tame hind repel one 
another, and bodies charged with electricity of different kinds 
attract each other. 

Resistance, may be conveniently regarded as that which 
opposes or resists the passage of the current. Most metals 
offer but small resistance, and are called good conductors, while 
wood, stone, silk and glass offer varying degrees of resistance. 
The unit of resistance is the ohm. 

Units of Electrical Measurement. 

It has been found essential to have certain units of measure- 
ment.-, solely adapted to express the force, resistance and cur- 
rent (so called) residing in electricity. 

Tie sc units were agreed upon at a Congress of Electricians, 
which met in Paris in 1881 and 1881 — the metric, or French 
system of notation was the one adopted in which 

A (j mn: iir- is equal to 15.432 grains (about 15A). 

A centimetre is equal to 0.3937 of an inch. 

The three principal factors, length, mass, and time are indi- 
dated by C standing for centimetre, G- for gramme, and S for 
second (of time); Ik nee the method is generally called the "C. 
G. S. system " of electrical notation, and the C. G. S. unit 
represents the work accomplished by the movement of a mass, 
equal to one gramme, through << space equal to one centimetre, 

r sin, ,,il Of lime. 

In representing the large numbers containing many cyphers, 
necessary for calculate ns in electrical quantities, the method 
was adopted of writing an exponent equal to the number of 
cyphers, thus: 10 8 is the equivalent of 100,000,000, because 8 
cyphers are added. 

An exponent or figure placed to the right of a letter or 
figure, above it, as H> . indicates that the number is to be mul- 
tiplied by itself, as in the example, 9 times. 

The Dyne, or absolute unit of force, is the force which, in 
one second, can impart a velocity of one centimetre per second 
to a mass of one gramme. 



2go Hand Book of Calculations, 

ELECTRICAL DATA AND DEFINITIONS. 

The Erg, or the absolute unit of work, is the work requisite 
to move a body one centimetre against a force of one dyne. 
There is an apparatus for measuring in ergs the work of an 
electric current. 

The Volt is the practical unit of electro-motive force, which 
would cause a current of one ampere to flow against the resis- 
tance of one ohm. One volt is equal to 10 8 absolute units. 
The volt is named from Volta, the original inventor of the 
primary battery. 

The Ohm is the unit of measure of resistance and is equal to 
10° 0. G. S. units and is approximately equal to the resistance 
of 129 yards of copper wire jVth of an inch in diameter, or, dif- 
ferently stated, such a resistance as would limit an electro 
motive force of one volt to a current of one ampere, or to one 
coulomb per second. The term is given in honor of Ohm, who 
discovered the law which governs electric resistance. 

The Ampere, the unit of electric current, or volume, is the 
ampere named after Ampere, who discovered and formulated 
the laws of electric currents. If an electro motive force of one 
volt be applied to send a current through a resistance of one 
ohm, the strength of current produced will be one ampere. 

An ampere per second is equal to one coulomb. An ampere 
may also be defined by the chemical decomposition the current 
can effect as measured by the quantity of hydrogen liberated, or 
metal deposited, shoivn in a device calleel the volt a metre. The 
latter really measures the coulombs and should properly be 
called a coulomb metre. 

The Coulomb is the unit of current quantity, considered 
with reference to time. It is named after Coulomb, to whom is 
due the first attempt at accuracy in electric science; it repre- 
sents the amount of electricity as would pass in one second in a 
circuit whose resistance is one ohm, under a electro motive force 
of one volt. 

The Farad is the electric unit of capacity, (named after 
Faraday). It represents the storage of one coulomb of electricity 
in a condenser. The Farad equals 10 ~ C. G. S. units of 
capacity. The micro farad represents one-millionth of a farad 
and =10 ~ 15 units of capacity. 



I land Book of Calculations. 2gi 



ELECTRICAL DATA AND DEFINITIONS. 

The W.Mi is tlie unit of electric power, named after James 
Watt, the inventor of the steam engine. The term volt-ampere 
means th< - the watt, as the latter is derived by multi- 

plying the two together, hence cue watt equals oue volt multi- 
plied into one ampere, which equals 10 7 C. G-. S. units of power. 

The Joule is the electric unit of heat. It represents the 
heat developed in a conductor by one watt in one second* 



Tn i aic Horse Power. 

The electric horst power, which is the equivalent of the 
mechanical horse power, is represented by 14»i watts — equal to 
<10 7 =7,400,000,000 0. <i. S. units of power. 

Electric Measuring Instruments. 

These are various in their principles of action and machine 
construction. By their use, quantity, cost and the commer- 
cial value of the electric fluid, so called, can 1m- determined to 
an* absolute certainty; the devices are known chiefly as volt- 
% and ammeter 8 ; voltmeters designating those which 
measure electro-motive force, the results being given in volts; 
and ammeters those which measure current strength, the 
- being given in ampere-. Ammeters are also called 
current meters and ampere meters. There are also instru- 
ments of special construction designated as electro-dynamo 
meters, coulomb meters and ohm meters. 

'I here are also instruments permanently placed in the circuit, 
like t 1 the boiler, as a guide to the engineer 

or attendant to indicate deviation- above 0T below a fixed E, 

M. I. tl 3 potential indicators. In the latter 

Inter is benl downwards over a scale 

which indicates to the right five volts above, and to the left, 

below, a Certain Standard indicated h\ zero; each of 

the numbers o] ... 10, 20, 30, 40, 50, indicating abonl a 

volt, each voll spa g subdivided into tenths. 



292 Hand Book of Calculations. 



Hints for the Engineer Relating to Electricity. 

The difficulties that beset the electrical engineer are chiefly 
internal and invisible; they are caused by leakage, undue 
resistance in the conductor, and bad joints which lead to waste 
of energy and the dangerous production of heat; bare and 
exposed conductors should always be within plain sight and as 
far out of reach as possible; the necessity cannot be too strongly 
urged for guarding against the presence of moisture, and the 
employment of skilled and experienced electricians in first 
erecting and supervising the work before turning it over to the 
engineer in charge of the whole plant. 

It is best to have a separate room for the dynamo and motive 
power, and if several dynamos are used it is equally important 
to have them all in one room. 

It is well for the engineer in charge to have an accurate and 
sensitive speed indicator, capable of showing even the slightest 
variations. 

A brush should never be lifted off the commutator while the 
dynamo is running. 

Every binding screw should be examined, and if necessary* 
tightened every day, as they are liable to be loosened by even a 
slight continuous jar cf the dynamo. 

The dynamo can be cleaned off, from the dust, etc., by 
means of a paint brush and a pair of bellows used daily. 
Shafts and pullies running near the dynamo must be prevented 
by means of shields from throwing oil on the dynamo and 
especially the commutator. 

It is advisable to run a new dynamo a few hours or even a 
day without any load, in order to have everything in proper 
working order before putting on the load, which should be done 
.gradually. 

The insulation of the coils of the dynamo should be practi- 
ally perfect. 



Hand Book of Calculations. 



-'93 



HINTS RELATING TO THE DYNAMO. 

All conductors in the dvnamoroom should be firmly support- 
ed, well insulated, conveniently arranged for inspection, and 
marked or numbered : the -f sign on electrical machines indi- 
cates the positive pole, and the — sign indicates the negative 
pole; for convenience sake it were well to assume that the 
electric current always flows from the positive pole of the gener- 
ator through the external circuit bach to the negative pole. 

The dynamo machine should be fixed in a dry place and it 
should he solidly set, the iron frame being properly insulated 
from the foundation, which is best done by a dry wood base 
plate. 

It should run steadily and evenly as any variation shows in 
the lights. 

It should not be exposed to dust or flyings. 

It should be kept perfectly clean and its bearings well oiled. 

Table. 
Weight ob Calendered Iron and Steel Shafting. 



Diameter 


Weight 


Diameter 


Weight 


in 


per foot 


in 


per foot 


In< 


(for iron.) 


Inches. 


(for iron.) 


| 


1.02 


2i 


13.25 




1.25 


2yV 


14.00 


I 


1.47 


2| 


14.76 




1.74 


O 7 

■"l if 


15.57 


i 


2.00 


2* 


16.37 




2.30 


2A 


17.20 


1 


2.61 


2f 


18.08 




2.96 


W 


18.91 


H 


3.31 


22 


19.79 


1A 


8.70 


2|;| 


20.71 


U 


4.09 


2i 


21 68 




4.60 




22.60 


if 


4.95 


3 


28 56 




5.41 




r 8* 


25.60 




5.89 




BA 


26 6jJ 


h 9 € 


8. W 




8 


27.65 




6.91 


'<• 


3| 


29 S'4 




7.45 


J 


V,. 


80.95 


If 


8.01 







82.07 


i 


8.60 


- 


8f 


84.40 


n 


9.20 







85.60 


Ml 


9.88 


M 




86.81 


2 


10.47 






89.81 


> 
■ 


11.15 






40.59 


n 


11.82 




i 


n 88 


* 


12.54 







2 94 



Haiid Book of Calculations. 



CHIMNEYS. 

The following table, calculated from approved formulae by 
Wm. Kent, M. E., is based on the supposition that a commer- 
cial horse-power requires — as an average — the consumption of 
five pounds of coal per hour : 

Sizes of Chimxeys, with Approximate Horse-Power of 

Boilers. 



3 *" 


Height of Chimneys and Commercial Horse Power. 


O Sh (O 

^ 2.2 

COCOh 


2 £ 

IPs 

w CZ3 


|8g 


50ft. 


6D 


70 


80 


90 


100 


110 


125 


150 


175 


200 


rfl 


18 


23 


25 


27 


















16 


0.97 


1 77 


2L 


35 


38 


41 


















19 


1.47 


2.41 


2t 


49 


54 


58 


62 
















22 


2.08 


3 14 


27 


65 


7° 


78 


83 
















24 


2.78 


3.98 


b) 


S4 


92 


100 


107 


113 














27 


3.58 


4.91 


a? 




115 


125 


133 


141 














30 


4.48 


5.94 


30 




141 


152 


163 


173 


182 












32 


5.47 


7.07 


39 






183 


196 


208 


219 












35 


6 57 


8.30 


42 






216 


231 


245 


258 


271 










38 


7.76 


9.62 


48 








311 


330 


348 


365 


389 








43 


10.44 


12.57 


54 










427 


449 


472 


503 


551 






48 


13.51 


15.90 


60 










536 


565 


593 


632 


692 


748 




54 


16.98 


19 64 


66 












694 


728 


776 


849 


918 


981 


59 


20.83 


23.76 


72 












835 


876 


934 


1023 


1105 


1181 


64 


25.08 


28.27 


78 














10 8 


1107 


1212 


1310 


1400 


70 


29.73 


33.18 


84 














1214 


1294 


1418 


1531 


1637 


75 


34.76 


38.48 


91) 
















1496 


1639 


1770 


1893 


80 


40.19 


44.18 


96 


















1876 


2027 


2167 


86 


46 01 


50.27 



Table of Size of Nails. 
The following table will show at a glance the length of the 
various sizes and the number of nails in a pound; they are 
rated i: 3-penny " up to "20-penny." The first column gives 
the name, the second the length in inches, and the third the 
number per pound: — 



3-pennv, 


1 i 


nch, 


557 nails 


per 


lb 


4 " 


H 


a 


353 


" 


i i 


a 


5 " 


If 


a 


232 


it 


a 


a 


6 " 


2 


a 


167 


a 


a 


a 


7 " 


n 


a 


141 


i ( 


a 


a 


8 " 


H 


a 


101 


« 


it 


a 


10 " 


n 


a 


98 


a 


a 


it 


12 " 


3 


a 


54 


a 


a 


a 


20 " 


H 


a 


34 


a 


a 


a 


Spikes 


4 


a 


16 


a 


a 


a 


a 


H 


a 


12 


a 


1 1 


a 


n 


5 


a 


10 


a 


a 


1 1 


it 


6 


a 


7 


" 


a 


a 


it 


7 


it 


5 


a 


a 


a 



Hand Book of Calculations. 



295 



Table Showing the Number of Days. 



From \ny 
Day OF 








TO J'HE 


9AMB 


Day OF NEXT. 


































Jan. 


Feb. 


Mar. 


Apr. 


May 


June 


July 


Aug. 

212 


Srpr. 


Oct, 


Nov. 


Dec. 


January 


965 


31 


59 


90 


120 


151 


181 


243 


273 


304 


334 


February 


334 


365 


■N 




89 


120 


150 


181 


212 


242 


273 


303 


March 


806 


337 


365 


31 


61 


92 


122 


1.-,.; 


184 


214 


215 


. 275 


April 


273 


306 


334 


365 


30 


61 


91 


122 


153 


183 


214 


244 


May 


245 


276 


304 


335 


365 


31 


til 


92 


123 


153 


181 


214 


June 


214 


245 


273 


304 


334 


3.55 


30 


61 


92 


122 


153 


183 


July 


184 


215 


243 


274 


3114 




365 


31 


(12 


92 


123 


153 


August 


153 


L84 


212 


243 


273 


304 


334 


365 


31 


• 


1)2 


122 


September 


122 


153 


181 


212 


242 


2 73 


303 


334 


3i 15 


30 


61 


91 


October 


92 


128 


151 


182 


212 


243 


273 


304 


335 


365 


31 


61 


November 


til 


92 


120 


152 


181 


212 


242 


273 


304 


3:54 


3H5 


30 


December 


31 


62 


90 


121 


151 


182 


212 


343 


274 


304 


335 


365 



AVlien February is included between the points of time, a day 
must be added in leap year. 

Example. 

1. How many days is it from the 10th day of March until 
the 16th day of October ? Now then : find March in first col- 
umn. Second, follow line of figures until column under 
" Oct."=214, add 6 days from the 10th to the 16th. Answer, 
220 days. 

2. How many working days from the 29th of July to the 
14th of September ? 

From the 29th of July to 29th of Sept. per table, 62 

Deduct the difference between 14th and 29th, 15 



Deduct each 7th day for Sunday, 



47 
7 



Answer, 40 days 

Note. 
The above table is of great value when used in connection 
with the Table of Wages, to be found on page 33, especially 
for verification of pay rolls, as well as for other- uses, both 
business and p. rsonal. 



2g6 



Hand Book of Calculations. 



TRANSMISSION OF POWER. 



Up to a certain limited distance, the useful effect of the 
energy created at a certain point can be best and most economi- 
cally transferred by belting of various materials, pullies and 
shafting. For greater distances there are four principal meth- 
ods in use for the transmission of power. These are 
1. Electricity. 

2. Water, (Hydraulic power.) 

3. Air, (Pneumatic power.) 
4. Wire Eope. 
On the basis of many experimental determinations the fol- 
lowing table has been computed, by Beringer of Germany, of 
representing the commercial efficiency of the different systems, 
under various conditions of distance and power, all the systems 
being supposed to be working to the best advantage. 

Table of Commercial Efficiency. 



Distance of 










transmission 


Electricity. 


Hydraulic. 


Pneumatic. 


Wire Rope. 


in feet. 










300 


.69 


.50 


.55 


.96 


1,500 


.68 


.50 


.55 


.93 


3.000 


.66 


.50 


.55 


.90 


15,000 


.60 


.40 


.50 


.60 


30,000 


.51 


.35 


.50 


.36 


60,000 


.32 


.20 


.40 


.13 



In these tables the maximum of comparison is 100 between 
the (four) systems. It will be seen that wire rope is the mosfc 
effective up to about three milts, beyond which electric and 
pneumatic transmission are most efficient. 

As the fundamental problems- of mechanical engineering are 
those relating to the generation, transmission and utilization 
of power, it were well for the engineer to study, with open-eyed 
attention, the problems now being- worked out in this direction, 
especially in the new field of electric transmission of power. 



Hand Book of Calculations, 2yj 



BELTING AND PULLEYS. 



Belts can be made of any flexible material, cloth, rubber, 
leather, and can be run in any way. at any angle, of any 
length, and any speed; 99 per cent, of all the power in the 
V. S. ifl transferred by belts and pullies, and yet all calcula- 
tions relating to them are subject to one fault, they are not 
positive, owing to the variatii n in the friction and consequent 
slippage of the belts on the pullies. 

Leather belting is the standard of comparison. 

The average strain under which leather will break has been 
found by many experiments to be 3,200 pounds per square inch 
of cross-section. A good quality of leather will sustain a some- 
what greater strain. In use on the pulleys, belts should not be 
subjected to a greater strain than one-eleventh their tensile 
strength, or about 290 pounds to the square inch oi cross-sec- 
tion. This will be about 55 pounds average strain for every 
imh in width of single belt three-sixteenths inch thick. The 
strain allowed for all widths of belting — single, light double, 
and heavy double — is in direct proportion to the thickness of 
the belt. 

The working adhesion of a belt to the pulley will be in pro- 
portion both to the number of square inches of belt contact 
with the surface of the smaller pulley, and also to the arc of 
the circumference of the pulley touched by the belt. This 
adhesion forms the basis of correct calculation in ascertaining 
the width of belt necessary to transmit a given horse-power. 
A single belt, three sixteenths inch thick, subjected to the 
strain we have given as a safe rule — 55 lbs. per inch in width — 
when touching one-half of the circumference of a tuned iron 
pulley, will adhere one-half pound per square inch of the sur- 
face contact: while if ii be a 1< at her-covered pulley, the belt 
will adhere two-thirds of a pound per 8quar6 inch of contact. 

If the belt touches hut one quarter of the circumference of the 
pulley, the adhesion ie only one-quarter pound to the square 
inch of contact with the iron pulley, and one-third pound per 
square inch on the leather-covered pulley. 



2g8 Hand Book of Calculations. 

Horse Power Transmitted by Leather Belts. 

In a single leather belt, not overstrained, a speed of 800 feet 
per minute for each inch in width is estimated to convey one 
hprse power. 

Examples. 
1. What power can be transmitted by a belt 7 inches wide 
travelling 1200 feet per minute ? 

1200x7 = 8400 
800)8400 



10^ Ans. 10i horse power. 
2. What power can be transmitted by a belt, single thickness, 
14 in. wide, running at a speed of 575 feet per minute ? 
2575X14=36050 
Divide 800)36050 • 



45 horse power, nearly. Ans. 
In a double leather belt, not overstrained, a speed of 550 feet 
for each inch in width will transmit 1 horse power. 

Example. 
What power will a double leather belt, 2-J inches wide, run- 
ning 4000 feet per minute transmit ? 

4000x2|=10000. Divide 10000 by 550=181 H. P. nearly. 
For calculating length of betting before putties are placed in 
position. 

Add together the diameters of the two pullies and multiply 
the sum by 3.14159. To half the result thus obtained add 
twice the distance from the centre of one pulley (or shaft) to 
the centre of the other pulley (or shaft). 

Example. 
Given the distance between centres of pullies 28 ft. 8 in.; 
diameter of pullies 52 inches and 46 inches.' What is the 
length of the belt ? Now then : 

Add the diameters 52+46 = 98 

Multiply 98 by 3.14159 = 307.87 

Divide this by 2 = 153.98 

Eeduce this to feet; 12 = 12.83 

Centres, 28' 8" X 2 = 57.33 



Add the last two together c- Ans. 70iV feet. 



Hand Book of Calculations. 299 

Rules fob Calculating Speed and Sizes of Pullies. 

When two pullies are working r connected by a belt, 

the one which communicates the motion is called the driver 
ami the one which receives 1: is railed the driven pulley. 

To find tin 1 size of the driving pulley: Multiply the diameter 
of the driven pulley by the number of revolutions it shall make 
and divide the product by the revolutions of the driver. The 

quotient will be the diameter of the driver. 

To find the number of revolutions of the Driven Pulley: .Mul- 
tiply the diameter of the driver by its number of revolutions, 
and divide by diameter of driven. The quotient will be the 
number of revolutions of the driven. 

diameter a ml revolutions of flic driver Ming given, to 
■find flic diameter of the driven that shall make a given number 

of revolutions: Multiply the diameter of the driver by its num- 
ber of revolutions, and divide the product by the number of 
revolutions of the driven. The quotient will be the diameter 
of the driven. 

Rules fob Calculating the Siz:: of Pullies for Ball 

Governors. 

To find the diameter of the governor shaft pulley. 

1. Multiply the number of the revolutions of the engine by 
tin- diameter of the engine shaft pulley, and divide the product 
by the number of the revolutions of the governor. 

•j. To find the <Iui>h<i>t of the engine shaft pulley. Multiply 
the number of revolution- of the governor, by the diameter of 
the governor shaft pulley, and divide the product by the number 
of revolutions of the engine. 

For finding the length of a roll of belting. 

Take the over all diameter, and add to it the diameter of the 
hoi.- in the centre of the roll; then divide the sum by two to 
find ///'■ mean diameter; — thie multiplied by 3.1416 (3jth) will 
give the circumference. Nex< multiply this by the number of 
ps^andthe result is obtained in inches, and bj dividing 
by 12 the length of th aed in f< 



joo Hand Book of Calculations. 

Useful Points Relating to Belting. 

The adhesion, one inch in width of the belt, has on the 
pulley is the number of pounds which each inch in width of 
belt is capable of raising or transmitting. Multiplying this by 
the velocity of the belt in feet per minute will give the total 
number of pounds each inch in width will raise or transmit one 
foot per minute. The answer is in foot pounds of which 
33,000 = 1 horse power. 

The thickness, as well as the width of belts, must be consid- 
ered; consequently, a double belt must be used where it is 
necessary to transmit a greater power than possible with a 
narrow belt. 

To increase the driving power of a belt, the pulleys may be 
enlarged in circumference, thus increasing the speed of the belt. 
This can often be done to advantage, provided that the speed, 
be not carried above the safe limit. 

The width of a belt needed depends on three conditions: 1st, 
the tension of the belt; 2d, the size of the smaller pulley and 
the proportion of the surface touched by the belt; 3d, the 
speed of the belt. 

The leather in a belt should be pliable, of fine, close fibre, 
solid in its appearance, and of smooth, polished surface. The 
character of the workmanship should also be considered. 

After the elastic limit of a leather belt is reached we may 
stretch it to the breaking point without getting it any tighter. 

Belts derive their power to transmit motion from the friction 
between the surface of the lelt and the pulley, and from noth- 
ing else, and are governed by the same laws as in friction 
between flat surfaces. 

The friction increases regularly with the pressure; the more 
elastic the surface the greater the friction. 

To obtain the greatest amount of power from belts, the 
pullies should be covered with leather; this will allow the belts 
to run very slack and give 25 per cent, more wear. 



I [and Book of Calculations. 



JOI 



Pitch of the Teeth of Wheels. 

The pitch of the teeth of wheels is the distance apart from 
centre to centre of the teeth, measured on the pitch circle. 

The pitch circle ox pitch line is the circle passing through 
the body of the teeth, which expresses the circumference of the 

wheel. 

In the annexed Fig. 
133, showing the halves 
of a wheel and a pinion 
in gear, A B is the line 
of centres, and C C and 
C C are the pitch circles 
touching at c ; the divis- 
ions A c and B c, of the 
line of centres, being the 
pitch-radii of the wheels. 
The arc of the pitch- 
circle, between 7> and /a 
is the pitch of the teeth, 
and it comprises a tooth 
and a space. 

In all calculations for 
speed of toothed gears, 
the estimates are based 
Fig. 133. upon the pitch line, the 

latter standing in the same place as the circumference of a 
pulley. 

THE PBOPELLER WHEEL. 

Thii is one of the mosl useful devices in the line of marine 
steam navigation ; it ranks as an invention, next to the ma- 
rine engine ; it is to the steam vessel, what the' exhaust- 
draught is to the locomotive, and its comparative proportions 
have been the subject of long and costly experiments. 

The propeller wheel consists of 2, 3, or 4 spiral or twisted 
blades, fastened to the main driving shaft of the vessel, where 
it comes through the Staffing boxes at the stern. 

The diameter of the propeller wheel is the diameter of the 
circle described by the extremities of the arms or blades. 




302 



Hand Book of Calculations. 



THE PROPELLER WHEEL. 

The pitch of the propeller, is the distance it would advance 
in a solid substance in one complete turn — like the turn of ^ 
screw. A true pitched propeller, is one whose blade has the 
same pitch throughout ; an increasing pitched propeller has 
blades whose pitch increases towards the tip, the decrease at 
the base being usually from 10 to 15 or 1(3 per cent, on the 
pitch at the tip. 

By "slip " is meant the difference between the actual advance 
of the propeller through the water, and the advance which would 
be made if the blades were working in a corresponding grooved 
solid body ; hence the apparent slip is the difference between 
the velocity of the ship and the velocity of the screw — the real 
slip is the difference between the speed of the screw and the 
speed of the water fed to the screw. The latter is frequently 
modified by the propeller wheel turning in the "eddy " or col- 
umn of water which follows the ship, instead of in the station- 
ary water of the sea. 

Method of Finding the Pitch of a Propeller. 
To find the pitch of the blade, take a lath from the stern 
post, and pnt it just touching the leading edge of the blade, 





U 

Fig. 134. 
and scrieve the lath and blade, shift it to the following edge at 
a point on the blade exactly opposite where the first pop was 
put, scrieve the blade and lath again. The distance between 






Hand Book of Calculations. 



303 



FINDING THE PITCH OF A PROPELLER. 

the marks on the lath Y (- is piece of the pitch ; on the blade, 
E F piece of the thread : square each of these, and subtract 
one result from the other ; now extract the square root of re- 
mainder, and this will give piece of circumference : take the 
circumference of the propeller at the place where the measure- 
ments were taken, and make a proportion as follows : 

As piece of circumference : whole circumference : : piece 
of the pitch : whole pitch. Answer. 

The above may he made clearer to some by the following : 
A c : C : : p : P 




: f r 

Fig. 135. 

A left-handed propeller is a screw with a left-handed thread, 
and a right-handed one has a right-handed thread ; therefore, 
a left-handed propeller to move the ship ahead goes from right 
to left, and a right-handed propeller turns from left to right : 
looking from tin- stern of tin- ship to the engine room and 
ngines passing the top centre. 

THE SLIDE VALVE. 

withstanding its extreme -implicit}' as a piece of mechan- 
ism, no pari of the engine is more puzzling to the average en- 
gineer in those questions relating to the lap and lead best to 

en it. tin- amount of. clearance, the proper point of cut- 

<■.. hence the following date is introduced to aid the 
student in acquiring the first points necessary to be known in 
practice. It is oearlj always a matter of years of study and 

ation before one becomes familiar with the subject. 



304 



Hand Book of Calculations. 



Putting an Engine on the Centre. 

Place the engine in a position where the piston will have very 
nearly completed its outward stroke, and opposite some point 
on the cross-head (as a corner); make a mark upon the guide, 
as shown at A in the accompanying figure. Against the rim 
of the fly -wheel place a pointer as at B and make a mark upon 
the wheel opposite this pointer when the cross-head is in line 
with the mark A upon the guide. Now turn the engine over 
the centre until the cross-head is again in the same position on 
its downward stroke. This will bring the crank as much 
below the centre as it was above before, being now in the posi- 
tion indicated by the dotted lines; and the point C on the fly- 
wheel will be opposite the pointer and should be marked. 
Divide the distance between B and accurately, and midway 
between them mark the point D. When D is brought opposite 
the point in the position which B occupies in the figure the 
engine will be upon the true centre. 




Fig. 136. 



to place the eccentric at right angles to the crank 
when the Piston is at either end of the stroke. 

1. Fasten a planed board at the eccentric side of the 
engine, in such a position that it will come under. the eccentric 
rod. 

2. Put on the straps and rod loosely. 

3. Then hold, or fasten a pencil to the rod, and have an as- 
sistant turn the eccentric once around, holding the pencil so it 
will mark the exact travel of the rod on the board. 



I laud Book of Calculations, 305 

TO GET ECCENTRIC AT RIGHT ANGLES. 

4. Find the centre of this line with a pair of dividers or a 
rule. 

5. Turn the eccentric up until the pencil conies to the cen- 
tre of line. 

G. Now fasten the eccentric so it wont slip. It is now at 
right angles to the crank. 

Directions for Setting the Slide Valve. 

Putting everything very accurately in the central position is 
the quickest and easiest way to set all valves. The rule is 
especially true of the common slide valve. The dotted line 
at A 1), Fig. 137, shows the centre of the valve and valve seat. 




Fig. 137. 



To set mi: Slide Valve. 

Study directions for putting the engine on the dead centre 
and observe Figs. L31 and L38 which represent a common 
slide valve and the eccentric. The direction of motion is 
shown V. 1 lit- arrows. 

1. Sel the crank on the forward dead centre. 

find the exact centre of the valve and mark it , with a 
fine line in such a manner that the line will show on top of 
tin- ralve ; also find the centre of all the parts, as shown at A, 
.. 131 : mark a fine line running up the side of the steam- 
so it can be -ecu above the ralve. 



jo6 



Hand Book* of Calculations. 



TO SET THE SLIDE VALVE. 

3. Then place the yalve over the parts, as shown in Fig. 137, 
and bring line on yalve and line on steam-chest, so they are 
together. This puts the valve in its central or neutral posi- 
tion. 

4. Put in the rod and connect it to rocker-arm ; plumb the 
rocker with a plumb-line and bob, so that the centre of eccen- 
tric rod pin will be cut by the line, and screw jamb nuts up to 
the valve with the fingers ; now fasten the valve so it can't 
move. Valve, rocker and eccentric are now in the neutral 
position, and temporarily fasten. 

5. The eccentric rod must now be brought into such a posi- 
tion that it will hook into the rocker-arm without moving it ; 
nowturn the eccentric the way the engine is to run until it has 
the proper lead or opening. If accurately done, the valve is 
properly set. To prove it put the engine on the other centre, 
and if the lead is the same, fasten everything. The valve is set. 




Fig. 138. 
Fig. 138 shows the position of eccentric on a direct acting 
engine when both engines (Figs. 137 and 138) are running in 
the direction indicated by the arrows. 

Emergency Bule for Setting Slide Valves. 
If the eccentric slips around the shaft, or any other accident 
throws the valve-gear out of position, then, 

1. Have some one roll the engine forward in the direction it 
runs until the crank is on the dead centre. 

2. Open the cylinder cocks at each end. 

3. Admit a small amount of steam into the steam-chest by 
opening the throttle slightly. 

4. Eoll the eccentric forward, in the direction the engine 
runs, until steam escapes from the cylinder cock at the end 
where the valve should begin to open. 



I land Hook of Calculations. 



3°1 



KMERGENCY RULE FOR SETTING SLIDE VALVES. 

5. Screw your eccentric fast to the shaft. 

6. Roll your crank around to the next centre, and ascertain 

in escapes at the same point, at the opposite end of the 
cylinder. If so, the valve is in position for service, until an 
opportunity occurs to open the steam-chest and examine the 
valve- gear. 

TO TAKE uATHS FROM THE VALVE AND VALVE SeA.T. 

This is done so that there may always be a permanent record 
kept of all sizes, &c, of the valve face. 
Fie. 130. 




Kg. 141. 

Wo will suppose that the valve is off and lying on its back 

on the engine room floor. Take a lath and lay \\ on the valve 

rad scrieve a mark al each edge of the valve faces (see 

e l 10), t! 11 is ;:t each Bteam and exhaust edge, next put 

a lath on the valve seal (Figure 139), and mark on it the 

. bars and exhausl port. Now lay 
the laths alongside each ether as in Figure ill: and see if 
both exhausl edges of the valve agree with the exhaust edges 
of the steam ports, if they do, well and good, it can be 

hy the-.- mark- how much Bteam lap, <xc, tlare is, as well 

making sure thai everything is in its proper central position. 



jo8 Hand Book of Calculations. 



Steam Expa^sio^. 

When steam is admitted to a cylinder during a portion of 
the stroke, then cut off, and expanded in the cylinder, upon 
the piston, for the remainder of the stroke, the pressure on 
the piston, during the period of admission, is or ought to be 
uniform, while the pressure during the period of expansion 
Qfalls as the piston advances and the steam expands. In en- 
gines in good working order, the expansion follows substan- 
tially the law of Boyle, or Mariotte, according to which the 
pressure falls in the inverse ratio of the expansion, the tem- 
perature remaining constant. 

Substantially, it is said, for the actual changes of j)ressure 
seldom follow the law exactly. The pressure usually falls 
more rapidly in the first portion of the expansion, and less 
rapidly in the last portion, than is indicated by the law ; and 
thus the final pressure may be, and it usually is, greater than 
that which would be deduced from the ratio of expansion. 

But the fullness of the expansion-curve shown on the indi- 
cator-diagram, near the end, compensates for the hollowness 
near the beginning ; and, sinking details, it is found that, 
practically, the area bounded by the curve is equal to that 
which would be bounded by a curve formed according to 
Mariotte's law. 

It is, therefore assumed, for purposes of illustration and the 
calculation of power, that the expansion of steam in the cylin- 
der takes place according to Mariotte's law. 

That is to say, if steam of 20 lbs. pressure per square inch 
be allowed to expand into double the space, the pressure will be 
10 lbs ; of triple, 6f lbs.; if 4 times, 5 lbs ; if 5 times, 4 lbs ; 
and so on. This theory would be literally correct did the tem- 
perature remain constant, but it is near enough to be consid- 
ered all that is ever required, and from its extreme simplicity, 
is universally adopted. 



I land Hook of Calculation*. jop 

STEAM EXPANSION. 

In other words, the product of the volume and pressure is 
always constant. All calculations must be made in absolute 
pressures; 100 pounds by the gauge equals approximately 115 
absolute. Starting with one volume at 115 pounds pressure 
and expanding to fill the second space, we should have two 
volumes at 57£ pounds pressure. An expansion to three 
volumes would reduce the pressure to 38J, and to four to 28f. 
The products of the volumes and pressures are always constant: 



Tolumea 




Pressures 




1 


X 


115 = 


115 


2 


X 


m = 


115 


3 


X 


3S£ = 


115 


4 


X 


28f = 


115 



It would take as many expansions to reduce a gauge pressure 
of 100 pounds to atmospheric pressure as 15 is contained in 

(.a ix From Steam Expansion. 



115- 



If the How of -team to an engine be cut off when the piston 
ha- made half its stroke, that is, if it is used expansively, it 
has been ascertained that the efficiency will be increased one 
and one seventh times beyond what it would have been if the 
. at half stroke had been released into the atmosphere, 
ami bo on, a- expressed in the following 

Table. 

Cutting <»tf ;u tl"« stroke, efficacy is increased 3.300 times. 

Cutting otf ,-it t tli<- stroke, efficacy is increased 3.000 times. 

Cutting off at '. the stroke, efficacy is increased 2.600 times. 

Catting offat ] the stroke, efficacy is increased 2.886 times. 

Cutting offal the Stroke, efficacy is increased 2.200 times. 

Catting off at \ the stroke, efficacy is increased 1.08 

Cutting ■• stroke, efficacy is increased 1.020 times. 

Catting offal efficacy is increased 1.690 times. 

Cutting offat lh< stroke, efficacy is increased 1.500 times. 

Cutting < , efficacy is increased 1.470 ti nes. 

iff al ' t h I 1.850 times. 

i tatting d 1.280 times. 



jio Hand Book of Calculations. 



THE STEAM ENGINE INDICATOR. 



The indicator is an instrument used for the purpose of 
recording the pressure of the steam in the cylinder, at all 
points of the stroke, as the piston moyes to and fro. This is 
done on a piece of paper secured to a revolving drum, by a 
pencil attached to the indicator piston. 

The indicator is said to have been invented by James Watt, 
but it was at first vastly inferior in finish and accuracy to the 
improved forms of the Eichards, Thompson, Crosby or Tabor 
indicators which are now largely in use; these are all substan- 
tially of the same construction and act upon the same principle. 

Each consists of a small cylinder accurately bored out and 
fitted with a piston capable of working in the cylinder with 
little or no friction and yet practically steam tight; the piston 
rod is attached to a pair of light levers at the end of one of 
which is carried a pencil designed to move on a nearly up and 
down line. 

The motion of the piston is controlled by a spring of known 
tension, several of which are furnished with each instrument; 
each spring is marked to show at what boiler pressure of steam 
it is to be used. The elasticity of the spring is such that each 
pound pressure on the piston causes the pencil to move a cer- 
tain fractional part of an inch. 

With Richard's Indicator there are 10 springs used; "No. 1 
will measure pressures from perfect vacuum, or 15 lbs. lelow 
the atmosphere to 10 lbs. above the atmosphere; No. 2 from 
- 15 lbs. "to -f 22i lbs.; No. 3 from - 15 lbs. to -f 35 lbs.; 
No. 4 from - 15 lbs. to + 47 lbs.; No. 5 from - 15 lbs. to -f 
60 lbs. ; and No. 6 from atmospheric pressure, or lbs. to -J- 
86 lbs. ; No. 7 from to 100 lbs. ; No. 8, to 125 lbs. ; No. 9. 
to 150 lbs.; and No. 10, to 175 lbs. 

Attached to the instrument is another hollow cylinder which 
has a diameter of about two inches, around which is placed the 
paper, the ends passing underneath a piece of slit brass, fitted 
so that the paper can be held firmly after being wound round. 

This cylinder is capable of a reciprocating or semi-rotative 
motion on its axis of such an extent that the extreme length of 
diagram may be 5 inches. 



Hand Book of C Calculations. 



3" 




THE INDICATOR. 

The part A is a steam 
cylinder containing a pis- 
ton B. The part of the 

cylinder below the piston 
L can be placed in connec- 
tion with either end of the 
engine cylinder. Above 
the piston is a spiral spring, 
C. D is a piston, to the 
head of which, E, it would 
be easy to fasten a pencil, 
the point of which could be 
made to press against a 
piece of paper, and by its 
upward and downward mo- 
tion register the steam 
Fig. 142. pressure beneath the piston. 

The diagram records the following facts and operations, viz. : 

1. The exact point of the stroke at which steam is admitted. 

2. The initial pressure of the steam in the cylinders, which 
being compared to the boiler pressure, shows us whether the 

m pipes and parages are of the necessary dimensions. 

3. The way in which the initial pressure is maintained or 
otherwise during the period of admission. 

•i. The point of cut-off. 

5. The pressure during tin- whole period of expansion. 
»i. The point cf release, i.e. when the exhaust is opened. 
T. The rapidity with which the exhaust takes place, as shown 
by the nature of the exhaust curve. 

he minimum hack pressure, which in a condensing 

- also the test of the perfection of the vacuum, and in 

►n-condensing engine shows whal the effect of the friction 

oftheexhausl pipesand p Is in addition to the unavoid- 

abh atmosphere. 

I n hen the e&hausl is closed. 
1". Th of compression. 

11. r I'h.- power which is being given "IT by the engine. 



3 12 



Hand Book of Calculations. 



THE INDICATOR. 



AW YB 




A 


C 




I 


> 
\ 




ll 




>■■ 


v t 


1 


■~-\v 


1 


, i 1 ! 1 i 



r 



i 



Fig. 148. 

In the figure, A B C D represents a card taken from a 
non-condensing engine. To find the average pressure from it, 
we divide the line VI) into equal spaces, as shown in the cut. 
Half way between every two vertical lines a dotted line is 
drawn, and the length of each dotted line is taken to repre- 
sent the average pressure betiveen the vertical lines on either 
side of it. Then the average of all the dotted lines will repre- 
sent the average pressure during one stroke of the engine. 

The most convenient method of measuring the dotted lines 
is by means of a long paper strip. Place the edge of it along 
the line V W and make a small mark opposite both V and W. 
Then move the paper strip so that the same edge lies along the 
line X Y and so that the mark that was opposite V now 
comes opposite Y. Then mark the point X, and so proceed. 
When the whole card has been measured in this way, the 
strip will look something like the lower cut of Fig. 143. 

A is the first mark and B is the last. The distance A B is 
next measured, and the result is divided by the number of 
dotted lines that are laid ofx along it. This gives the average 
length of the dotted lines, and thereforeit represents the aver- 
age pressure during the stroke of the engine. 

The length of the indicator diagram, multiplied by the aver- 
age height of it, will give the area of it ; so that after having 
found the average height (or width) of it we can find the area 
of it if we wish to. On the other hand, if we know the area 
of the card, we can find the average width of it by dividing the. 
area by the length. 



Hand Book of Calculations. 



3*3 



To work out an I\i»i< atok Card for a Low Pressure 

Engine. 

1. Divide the Card into ten eqnal parts by lines perpendicu- 
lar to the atmospheric line. 

'I. Measure the width of the card at the middle of every one 
of these parts on the scale. 

3. Add these measurements together and divide the sum by 
ten. the result is the mean pressure on the piston per square 
inch throughout the stroke. 

What is the mean pressure throughout the stroke in the fol- 
lowing : — 



Scale -ie of an inch 






! : 


ij • 


lq 


N. ' 


<n : 


<* 


*l : 


n : 


M 



Fig. 144. 

Answer, 18*8 lbs. per square inch. 
If the diameter of the pair of cylinders from which the above 
diagram was taken was 52", the length of stroke 42", the rev- 
olutions per minute 41, the steam gauge showing 15 lbs., the 
vacuum gauge 21 '. what is the indicated horse power ? 

In finding the I. II. P. the heights of steam and vacuum 
gauges and the barometer arc not needed, but they are gener- 
ally noted on the paper on which the diagram is drawn. 

The pressure shown by the steam gauge is useful to the en- 
gineer, because it enables him to see what is the loss of pressure 
ien the boiler and the cylinder ; for Instance, in the dia- 
gram in the full steam line is 12 lbs. above the atmospheric 
line, and the steam gauge showed 15 lbs. ; therefore there was 
of 3 lbs. ; but for the purpose of finding the EL 1'.. we 
only require to know the mean pressure throughout the stroke; 
in our diagram it ie 18*8 lbs. Now then, 

= 34 r.235 feel l cylinder. 
2 






694. tTO f< :■ the pair. 



3*4 



Hand Book of Calculations. 



Examples. 
This cut represents a diagram taken by C. W. Simmons, 
from a 12"x24 // automatic engine, speed, 126 revolutions, scale 
40 lbs. boiler pressure T8.6 lbs., M. E. P. 46.41 lbs., H. P. 80 T y ¥ 




Fig. 145. 

Explanation : A. A. — Atmospheric line, which is drawn 
with the atmosphere admitted to both sides of the indicator 
piston. B. C.— Admission line. 0. D.— Steam line. P.— 
Point of cut-off. B. E.— Expansion curve. E.— Exhaust. 
E. F. — Exhaust line. F. G. Counter pressure line. G.— 
Point of exhaust closure. G. B.— Compression curve, II. H. 
H. — Lines drawn for the purpose of ascertaining the average 
pressure. 

Example 2. — Find the mean effective pressure in the cylinder 
of a condensing steam engine when the pressure of steam on 
admission is 80 lbs. absolute, cut off at one-fourth of the stroke. 
Back pressure 3 lbs. per square inch. 



bu 


-)S A 


\ 


















60 




1 


\ 














W 








I 


e 










20- 




' CLt;mo 


•hAc/ric 


i 

Jujn*. 







• 


i 


T~~" 






I 1 i 




t 
i 

n1/ 


« 


<4> 




II 

4. 


8 


,» 








1 


















"J 


>> * 
>• > 




3 




3 




53 




^ 


| 



Fig. 146. 



Hand Hook of Calculations. Jff 

Principal CAUSES A.FFECTING THE FORMS OF DIAGRAMS. 

1. The friction of the steam pipes and ports. 

2. The variable size of the opening of the sfceam ports as 
caused by the gradual motion of the slide valve. 

3. The action of the sides of the cylinder in causing conden- 
sation and partial re-evaporation of some of the entering steam. 

4. The strain contained in the clearance spaces which affects 
the curve of expansion. 

5. The gradual opening of the exhaust port, which makes it 
necessary t<> release the steam too early in the stroke. 

6. The friction of the exhaust passages. 

7. The momentum of the moving parts which combined 
with cause 4- and also with the unavoidable nature of the simple 
slide valve driven by an eccentric, renders a curve of compress- 
ion necessary. 

77/'' only absolute information the diagram conveys, what- 
ever its form, is the pressure in the cylinder of the engine. 
All the other information to be had from it comes through, a 
<s of reasoning based upon experience and observation. 
In order, als >, that the diagram shall be correct, it is essen- 
tial, first, that the motion of the drum and paper shall coincide 
exactly with that of the engine piston, and second, that the 
motion of the pencil shall correspond with the other motions 
ibed. 
It is beyond the province of this work to much more than 
explain and illustrate the calculating of the diagram after it 
shall have been obtained; to do the latter accurately, LI is 
Mat skill which alone comes with practice. 

X < > ' 
The passage of Bteam into either end of a cylinder may he 
distinctly heard bj puttings rule, a piece of wood or a piece of 
iron betw< q the teeth, a> pping the ears with the fingers, and 
putting ' end of the piece of wood or iron against the 

cylinder. Both ends t the cylinder maj be "heard " in turn; 
both pill sted in the same manner for lost 

motion, . m;i\ be t ried lor end mol ion in 

tin- \ ion. 



ji6 Hand Book of Calculations, 

BUSINESS FORMS FOR ENGINEERS. 

When the engineer comes in contact with men and the busi- 
ness of life, he needs another set of calculations from those he 
uses in dealing with the natural and mechanical forces. The 
following are designed to aid him in this direction, reference 
being made to the table of wages, page 33, and the table of 
days, page 295. 

How to Make Figubes. 

Clearly made figures, neatly placed in their appropriate posi- 
tion for adding, subtracting and dividing, etc., are most agree- 
able to the eye, both of the maker and reader. It is worth 
much effort to acquire a ready faculty in this direction. 

Fac Simile of the Author's Figures. 

/ & 3 <¥-<$'■$' f //• 6y 




In this system of forming the numerals, it will be observed 
the hcighth of the whole nine digits is nearly uniform i. e. the 
o which forms part of the 6 and the 9 is made as long as the 
cypher when placed alone ; that the top of the 7 is lined with 
the top of the 8, the stem coming as far down as does the stem 
of the 9, while the stem of the 6 extends as far upwards as 
does that of the others downward. 

The above specimen of toritien figures is presented as a 
model for practice in writing down the numerals. 



Hand Book of Calculations. jiy 



Bills. 

A Hill is a formal written statement of goods sold or services 
rendered^ or both. 

Every bill contains : 1. A date ; 2. The debtor's name ; 3. 
Tin- creditor's name, after the words Bought of, or between 
the word- To ami Dr.; 4. The statement of goods sold or of 
services rendered, or of both, with prices and amounts. 

When tin.' bill is for goods, the expression Bought of, is gen- 
erally employed : when for services, or for both goods and ser- 
vices, the word Dr. is used. When this word is used, each 
item is usually preceded by the word To. 

The bill is made by the creditor, and by him presented to 
the debtor for payment. 

There arc two parties to every bill, a creditor and a debtor. 

The Debtor is the person who receives the goods or services 
or both, and who therefore owes for them. 

The Creditor is the person who supplies the goods or renders 
the services, or does both. 

When the debtor pays the bill, the creditor receipts it, or ac- 
knowledges the payment, by writing his name on the bill, 
under the words Received Payment. 

Where to use Capital Letters. 

1. Begin with a capital the first word of every sentence, 

2. Begin with a capita] every proper name, 

■ >. Begin with a capita! titles of honor <nnl /■/'*/><■<■/. 

4. Begin with a capital all appellations of God, 

'». Begin with ;i capital the days of tin 1 week mid months 
of (he year. 

*;. Writ*,' with capitals the pronoun I (nut the interjec- 
tion o. 

!'. Begin with a capital the words North, South, East, and 
West, when they denote a - ction of country. 



ji8 Hand Book of Calculations. 

Business and Law Points. 

The law compels no one to do impossibilities. Notes bear 
interest only when so worded. Agents are responsible to their 
principals for errors. Ignorance of the law excuses no one. 
Principals are responsible for the acts of their agents. Con- 
tracts made on Sunday cannot be enforced. A contract made 
with a minor is invalid. A contract made with a lunatic is 
void. It is a fraud to conceal a fraud. Signatures made with 
a lead-pencil are good in law. Each individual in a partner- 
ship is responsible for the whole amount of the debts of the 
firm. A receipt for money paid is not legally conclusive. An 
agreement without consideration is void. A note given by a 
minor is void. The acts of one partner bind all the others. It 
is not legally necessary to say on a note ''for value received." 
A note drawn on Sunday is void. 

Lettees. 

Every letter consists of six parts, as follows : 1. Date ; 2. 
Address ; 3. Salutation ; 4. Body ; 5. Subscription ; 6. Su- 
perscription. 

The Date is a statement of the place and time of writing. 

The Address is the name of the person to whom the letter is 
written. 

The Salutation is the brief expression of greeting which im- 
mediately precedes the body of the letter. 

The Body of the letter is the communication itself. It con- 
sists of divisions, as there are subjects discussed. 

The Subscription consists of expressions of regard or compli- 
ment with which the letter closes, and the signature. 

The Superscription is the full and particular address written 
on the envelope. 

Daily Memoeandum Book. 
Each engineer should keep a book of convenient size ivherein 
to write down, from day to day, all items of business engage- 
ments, neiv mechanical facts, rides and processes. This corre- 
sponds to a journal in a set of books, and the items, if written 
seriatim, i. e. one after another, are evidence in a court of 
law. 



Hand Book of Calculations. 



3*9 



< lsh Re< i:ivn> and Paii> Out. 

Money being the nerve centre of modern industry, it were 
well for the engineer to keep an accurate record of the smallest 
sum which passes through his hands for any purpose. This is 
best done in a separate book or memorandum, which may be 
called the cash-book. This can be ruled into one wide 
space and three small spaces — as shown in the example. 

In these four Bpaces there should be kept. 1. The date. 2. 
The particulars of the cash transaction. 3. The cash received, 
and 4. Cash paid out. 

Example. 



D.v: 




Receiv'd 


Paid out. 


1890. 






% |Ct8 


$ ;cts 


Jan. 


1 


( 'ash on hand in Savings Bank 

currency and silver. 


100 00 
4280 








Pai'i mo' month's rent to W. J. Jones 






40 00 




10 


Rec'il for sub-rent of W. Will unit* . . . 

Paid house expenses to daU from Jan. 1 

et for subscript' u to paper for 1890 


18 


00 


x \ 


40 
00 




11 


" dues to Engineer's Association 
to date 






2 


58 




Recfd for two week's wages M. Mfg. 








Co., $22.00 


44 


00 








15 


•• from firm to buy files, etc 

Paid for files, 75c:; waste, L.25; mon- 


4 


00 










key wrench, 1 .00, 






3 00 




Paid for gaskets for boiler II. holt \ 


1 00 




20 


•• plumper for work at houst .... 
•• for stove $15; 1 bbl. offloi $< 






3 00 
n 00 




%A 


•• payment on house and tot (total 












paid $000) 






25 


00 






Rrc'd interest on balanct in Savings 














Bank 


4 


80 






25 


two week's wages M. Mfg Co. <" 






$22 


44 


(Ml 






31 


Paid family expt nst - 1 > date 






23 


fin 




•• personal axp month. 










daily /injur, car fan , ett 






1 00 




toSavingsBank,{ interest added) 






4M> 


Oh hand 






il2'22 




1 




60 


251 60 



in- model the second month (Feb. L890) would begin 
with Cash on hand. $1 L2.22 and then the new month's entries 

■ ll"\\ in their order. 



J20 



Hand Book of Calculations. 



BOOK OF RECORDS. 























d 


















< 


















J; c cc 

d ^,Q 


















O . 
«'^ 

a© 
Eh 


















It 

<u o 


















^ 

^M 


















.2 a? 

Sao © 

a 


















g 60 ■ 

ce u d 
^ a 
















I 


CO 

p s 

o p 1 








1 


, 






3<D 

cy be 

^ d 




















o 

3 


P 




H 

EH 


3 


Eh' 
W 


i 


I 


.Cl 

d 

















Recokds. 

A general record or 
log book should be 
kept in which to 
record each time the 
engine is started and 
stopped. 

There also should 
be noted the average 
steam pressure and 
vacuum for the day, 
the temperature of 
engine room, injec- 
tion and discharge of 
water. 

Pasted into this 
same record should be 
indicated diagrams 
worked up for each 
cylinder, care being 
taken that they rep- 
resent, as nearly as 
possible, the average 
performance for the 
day. 

A form of a dia- 
gram is here present- 
ed, which the engi- 
neer can follow in 
ruling a book to suit 
his own steam plant, 
modifying it so as to 
fit the surrounding 
circumstances. 



INDEX 



,\X1) USEFUL DEFINITIONS. 



Abbreviation*, used in geometry, 
129. 

A. B. Indicates "Above atmosphere/ 1 

Acute Angled Triangle, 133. 

Acre, aliquot parts of, 38. 

Acute Angle, 131. 

Adlabatlc, as applied to an expansion 
curve, means that it correctly rep- 
resents at all points tht pressure due 
both to the volume and the temper- 
ature. 

Addition, 19. 
Table, 18. 
Rule, 19. 

Rule for proving, 19. Kxamples,19-23 
Bignof, 12. 1'.'. 

Air, composition of, etc., 197. 

cubic feet of, in a chimney, 248. 

Algebra, advantages of, 223. 
elements of, 821, 

algebraic forms, 223. 

Alcohol, specific u r r;i'. ity, 218. 

A I i<i ii <>i part-, definition, 38. 
Tab 

Alloy*, melting point - 

Altitude, of a polygon, 134. 

Ammeters, 291. 

Amount, 19. 

Ampere, the, 290. 

Angle, 

abbreviation, 129. 
Antieedent, of a ratio, 160. 
Antimony, melting point, 280. 

specific gravity, 217, 
Apex, i:;i. 

A pot heearie* weight, table, 34. 

Arable uotat ion, 17. 

1 nil*, table, 1 1 » 126, 

tr< ;i of boiler tube*, rule to com- 

put< . 
Arithmetic. 13. 
Arithmetical I'ormiilaH, 12. 
-. 12. 
p r ogr e s si on, ill, l7-">. 

Armature, the, 286. 



As St., Abbreviation for Assistant. 

Atom*, 4li. 

Are, of a circle, 134. 

Avoirdupois Weight, table, 31. 

Axis, the straight line, real or imagi- 
nary, passing through a body on 
which it revolves or may revolve. 

A \ in ms, 128. 
examples, 130. 
abbreviation of, 129. 
. Abbreviation for At. 

Barometer, 282. 
weather, 283. 

Base of a polygon, 134. 

Belting & pit Hie*, 297. 

Belting, use Cul points relating to, 300. 

Bismuth, melting point, 280. 

Bill*, description, 317. 

Boiling point, of fresh water, 283. 
of salt water, 283. 

Boiler tube*, rule to compute area 
of, 239. 

Boyle* law of expansion, 308. 

Bra**, weight of, 36. 
specific gravity, 217. 

Bru*he*, colled ing, 286. 

Bronze, melting point, 280. 

Business forms, lor engineers, 316. 
point-, 318. 

C. , symbol of, 289. 

Calendar of months and days, 85. 

Calculus. A term applied to various 
brandies of algebraical analysis— 
a inane], of mathemal lea little used 
in practical mechanics. 

< a naila Mom-) , 817. 

< a mile pow er, 269. 

Cancellation, 108. 

D of, 10H. 

rules, 109. 

examples, 108, 109,110. 

Carbonic add, mell Ing point, 280. 
Cardinal Numbers are those which 

I bie amount of units, as 1,2, 
8, t,:., 8, 7,8, 9, 0. 
Center, putting engine on, 304. 



3 22 



Index and Useful Definitions. 



Cash account, form of, 319. 

received and paid out, 319. 
Centimetre, 39, 389. 

symbol of, 289. 
Centigrade Thermometer, 159. 
Centrifugal force is that by which 

all bodies moving around another 

body in a curve, tend to fly off from 

the axis of their motion. 
Centripetal is that which draws, or 

impels a body toward some point as 

a center. 
C. G. S., symbol of, 289. 
Chains, iron, 81. 

proportion, 81. 

example, 81. 

table of strength, 82, 

comparative cost. 
Chemistry, 46. 
Chemical affinity, 47. 
Chimneys, description, 248. 

table of proportion, 194. 

cubic feet in, rule, 248. 
Closed circuit, 287. 
Classifications of steam engines, 249. 
Circle, 134. 

sign of, 129. 

rule to find circumference, 93. 

rule to find area, 94. 
Circular inches, 196. 

rule and examples, 196, 197. 
Circumference, 134. 
Circumferences, table, 114—126. 
Circumference of Circle, sign, 32. 
Circular measure, table, 32. 
Circuit, 287. 

Circulating decimal, 227. 
Co., meaning of, in trigonometry, 137. 
Coal, specific gravity, 217. 
C. O. I>. Cash on Delivery. 
Coefficient, 222. 
C/o, Abbreviation for Care of. 
Cohesion, 228. 

Cohesion is that quality of the parti- 
cles of a body which causes them to 

resist being torn apart. 
Color of hot metals, table, 159. 
Combustion, power of, 47. 
Coulomb, 290. 
Commutator, 286. 
Compression, or direct thrust, 228. 
Compound numbers, 42. 
Complement, of an arc, 137. 
Compound engine, 250. 

rule to find horse power, 167. 
Concave. Hollow, arched and round 

as the inner surface of a spherical 

body opposed to convex. 



Condensing Engines, 166 & 250. 

rule to find II. P., 1G6. 

to work out a card, 315. 
Conduction of Heat, 152. 
Conducting power of metals, 

table, 153. 
Cone frustrum of, 105. 

to find cubic contents, 105. 

example, 105. 
Connecting rods, 250. 
Continued Fraction, 227. 
Contents of solids, 103. 
Contents of geometric solids, tc 

find rule, 135. 
Convection of heat, 153. 
Convex. Rising or swelling on the 

exterior surface into a round form; 

opposed to concave which expresses 

a round form of the interior surface. 
Consequent, of ratio, 150. 
Copper, meting point, 280. 

tensile strength of, 231. 
Copper Wire, tensile strength of, 231. 
Corollary, 129. 

abbreviation, 129. 
Cord of Wood, 36. 
Cosine, of an arc, 137. 

abbreviation, 137. 
Cosecant, of an arc, 137. 
Cotangent, of an arc, 137. 
Coulomb Meters, 291. 
Counting Measure, table, 35. 
Counters, for steam engine, 268. 
Cranks, of steam engine, 259. 
Crank pin, of steam engine, 269. 
Crank shaft, description, 262. 
Crossheads of steam engine, 258. 
Crushing strength, of material, 

128. 
Cube root, 194. 
Cubit, 39. 

Cubes, table of, 275—279. 
Cubic inches of Bushels & gal- 
lons, 36. 
Cubic or solid measure, table, 34. 
C wt. , abbreviate for Hundred weight. 
Cylinder, a solid, rule to find cubic 

contents, 104. 

example to illstrate rule, 104. 

rule to find area, 255. 

example to illuustrate rule, 102. 
Cylinder, steam boiler, rule and Ex 

of H. P., 234. 
Cylinder, of steam engine, 152. 

rule for H. P., 254. 

rule to find capacity, 253. 
Daniels' pyrometer, 158. 
Data of Electricity, 286, 



Index and Useful Definitions. 



3 2 3 



Data. Things given or admitted; 
quantities, principles, or facts given, 
known, or admitted, by which to 
find things or results as y el unknown 

Dead load, (A), on a structure, is one 
that is put on by imperceptible de- 
grees, and thai remains stead\; 
Mich as the weight of a boiler or an 
engine on their foundations; op- 
posed to live load. 

Decimal, acirculating, 227. 

Decimetre, 39. 

Decimal system, 17. 

Decimals si, 84. 

multiplication of, 86, 87, 
subtraction of, 86. 
examples of, M & 85. 
addition of, 85, 86. 
division of, 88,89,90, ML. 
system of notation, 17. 

Decree, 82, 130. 
_ 1 1 of, l~.». 

Denominator, 106. 

Diagram, principal causes affecting, 
8l0. 

Diagonals, 134. 

Diameter, 134. 

Diameters, table, 114-126. 

Diameter of engine shaft pulley, 
rule to get, 24. 

Diameter of propeller wheel, 301. 

Difference, arithmetical definition, 12 

Difference, symbol of, 129. 

Disintegration of steam. 199. 

Division, sign of, 12. 

Divide ml, 28. 

Division, Definition. 28. 

examples and rules, 28, 2ft, 30. 
tab; 

Divisor, l 

Dodecahedron, a figure, 136. 
Dr., 317. 

Draught of chimneys, rule for cal- 
culating, 248. 
Dry measure, table, 32 
Ductile. This means the capacity for 

bending. 
Ductility, 230. 

D] IK. 

Dj it a mics i-t bat branch of mechanics 

which treats of bodies in motion; 

opposed t'» -'"' 
Dynamo, tb< . 
I. a. . . bbreviation for Each. 
Earth's specific gravity, 217. 

i;<cent ric-hook, 

Bet l ntric, rule for putting, at ritfht 
angle 



Eccentric rod, 262. 

Eccentrics, 261. 

Eccentric strap. 262 

■elasticity is that quality of a body 
which enables it to return to its 
original position after having been 
stretched. 

Electrical energy, 286. 
Electricity, 4"), 47, 285. 
Electrical data, 286. 

positive and negative, 289. 

units of, 14. 

Elect ricit y, as an industrial agent, 285 

Electric measuring instruments, 291. 

Elevation* A view or representation 
of an object or machine, drawn to a 
geometrical scale, one having no 
vanishing point— a side view of a 
drawing. 

Elementary or simple body, 4i>. 

Ellipse, rule to find area of, 97. 

Emergency rule for setting slide 
valve, 306. 

E. M. F., 287. 

Engineer's records, form for, 320. 

Engineer's signal code, 274. 

Engines, Primary, 49. 

Engine on centre, putting, 304. 

Equality, si^n of, 12, 129, 222. 

Equiangular triangle, 132. 

Equilateral triangle, 132. 

Erg, 290. 

Estimate means to compute, to calcu- 
late, to reckon. 

Evolution, 191. 

Exhaust steam, value of, 207. 

Exponent, 289. 

Expansion of steam, 207, 208. 
Table, 309. 

Extensibility, 230. 

Extremes of a proportion, 150. 

Face of safety valve, 240. 

Factor, 222. 

Factor of safety of steam boiler, 236. 

Factors, 26. 

Fathom, :>'.h 

Farad, 290. 

Fahrenheit Thermometer, 155. 

Fatigue Of metals. Note, 232. 

Figures, description, 16. 

How to read I hem, 16. 

How to make them, 816. 
Figures of four Bides, L88. 

Flexion or transverse strain, 220. 

Flexure. This is thepoinl in a dia- 
gram ;it. which thecut-olf closesand 
the expansion curve begins. 

Fluid, 10. 



3H 



index ctncl Useful Definitions. 



Flywheels, 267. 
Foot-pound, 160. 
Force, 160, 254. 
Formulas, Algebraic, 223. 
Formula, for determining strength 
of boilers, 226. 

for determining H. P. of engine, 227. 
Formulas, Examples, 224 and 225. 
Fractional cents, Note, 44. 
Fractions, vulgar or common, 106-108. 

A proper, 106. 

An improper, 106. 

Addition of, 110-112. 

Subtraction of, 112. 

Multiplication of, 112 and 113. 

Division of, 113. 

A continued, 227. 
Franc, (A,) 37. 
Freezing- point of water, 176. 
Frencli money, Table, 7. 
Friction of belting, 300. 
Fulcrum, 50. 
O., symbol of, 289. 
Crab, or eccentric hook, 262. 
Gain from steam expansion, 309. 
Galvanism, 47. 
Gas, illuminating, 269. 

Points important to know, 271-272. 

Meter, 271. 

Meter, the index, 271. 

Meter, how to read, 271. 
Geometrical magnitudes, 128. 
Geometrical problems, 138 to 148. 

Solids, 135. 
Geometry, 127. 
German money, 37. 
Girtli seams are the seams which 

pass around the body of the boiler; 

these are usually single riveted. 
Glass, tensile strength of, 231. 
Governors, 265. 
Gold, specific gravity, 217. 

Melting point, 280. 
Government land, measurement of, 

36. 
Grate-bars, rule to compute area, 238. 
Grate surface. This means the total 

square feet in the grate-bars, as 

they are arranged in the furnace for 

firing upon. 
Gravity, 212. 

Table, 212. 

Specific, of bodies, 215, 
Rules and examples, 212-214. 
Gramme, 289. 
Gravitation, 47. 
Hair's breadth, 39. 
Hand, (A,) 39. 



Hearing through the teeth, 307. 
Heat, 151. 

(or thermal) units in 1 lb. of water, 

Table, 208. 

True nature of, 154. 

The unit of, 14. 
Heat, internal, of the earth, 47. 
Heat, table of, produced by different 

lights, 271. 
Heating surface of steam boilers, 172 
Hence, or therefore, sign of, 222. 
Heptagon, 133. 
Hexagon, 133. 
Hexahedron, 135. 
High pressure engines, 249. 
Hints relating lo the dynamo, 293. 

Relating to electric it y, 292. 
Homogeneous. This word as ap- 
plied to boiler plates means even 

grained. In steel plates there are no 

layers of fibres, but the metal is as 

strong one way as another. 
Horizontal means level. 
Horizon. An imaginary circle touch- 
ing the earth and bounded by the 

line in which the earth and sky 

seem to meet. 
Horse power of the steam engine, 160. 

Rule second and examples, 163. 

Rule three, 165. 

Rule four and examples, 166. 

Examples and rules, of steam boiler, 
172, 173. 

Note relating to condensing engines, 
166. 

Note relating to steam fire engine, 
170. 

Rule for, of locomotives, 168. 

Rule for, of steam boiler, 172. 

Rule for steam fire engine, 170. 

Of a vertical boiler, 224. 

Rule for, of condensing engines, 166. 

Of locomotive boiler, 234. 

Rule and example for power of com- 
pound engine, 167. 

Of cylinder boiler, 234. 

Colburn's rule of power for the loco- 
motive, 170. 

Effective, 161. 

Nominal, 161. 

The electric, 291. 

Indicated, 161. 

Rule for calculating, 162. 

Transmitted by the leather belt, 298. 

Notes, 161 and 164. 
H. P. means Horse Power. 
H. F. Cyl. Indicates the High Press- 
ure Cylinder. 



Index and Useful Definitions. 



325 



Hydraulics, 17t>. 

Hydrogen. l!>7. 

Hydrometer, 210. 

Hydrometer, 

Hydrostatic*, 4"). 

Hydrostatics, 170. 
Abbreviation of, 12?>. 

Hypothesis, 12S. 

ice, melting point, 280. 

Ico*ahcd 1*011, 1 :>.">. 

I. II. P. Indicated Horse Power. 

Incandescence. White or glowing 
with heat 

Inch, Circular, 196. 

Inclined plane, <*i. 
Rules, 68, 89. 
Examples, 68, r>7. 

Indicator, Bteam engine, 310. 

Indicator, besl method of measuring 
diagrams, 313. 

Indicator*, potential. 291. 

Injectors, 209, 210, 211. 
Note, 211. 

Inst. Abbreviation for This month. 

Insulators of electricity, 288, 

Integers are whole numbers. 

Inverse means inverted, opposed to 
din it. 

Involution. 193. 

Isosceles triangle, 132. 

Isosthernue, The, as applied loan 
expansion curve, means that such a 
curve represents correctly the ex- 
pansion or compression of the steam 
when the temperature is uniform. 

Iron, weight of, 36. 
Tensile Btrength, 231. 
Melting point, 280. 
Bpecific gravity, 217. 

Jorlc, The, 291. 

Jorles equivalent of heat, 1">4. 

journals, ~m. 

Knot, 89. 

Lactometer, 284. 

Law points, 318. 

I. cad, melting point, 280, 
Bpecific gravity, 217. 

Leather belting, power transmitted 
by, . 

Lemma, 

Length, the unit of, \i 

Length of roll of belting, rule for get- 
t og, 899. 

Length, 

Length of belting, rule for calculat- 
ing, 
Letters, what the] conslsl 1 



Lover. A lifter. This is the first 
meaning of this often used word. 

Lever, 4i> and GO. 
First kind, 60. 
Second kind, 50. 
Third kind, illustration, ">1. 
Exam jdes, 61. 

Rules and examples, 52, 5:}, 54, 56, 56. 
General rule, 51. 
Special note, 51. 

Light, table of cost, 270. 

Rule for measurement, 270. 

I. i in it of Btrength in material, 230. 

Line, 12S and 129. 

Liquid measure, Table, 32. 

Live load. One that is put on sud- 
denly, or is accompanied with vibra- 
tions, the force exerted by the con- 
necting rod of an engine is a live load 

Locomotive boiler, rule and exam- 
ple of H.P.,235. 

Locomotive engines, 108 and 169. 
Colburn's rule of power, 

Longitudinal seams are the riveted 
lints which run in the same parallel 
direction with the boiler, i. e., from 
the front to the back; these seams 
are usually double riveted. 

Long ton for coal, 35. 

Long or linear measure, table, 31. 

Low pressure engine, 240. 

Low pressure engine, to work out 
a card, ;H4. 

L. P. Cyl. Indicates the Low Pressure 
Cylinder. 

Luminosity at high temperatures, 159 

machinery, 49. 

Machine*. 48. 

Maiiiielism, 47. 

magnets, field, 286. 

magnitude, 13. 

.llarine engines, 351. 

marlotle's law of expansion of gases, 

308. 
fliirk, :}7. 

Ma**, 251. 

mathematics. The science of quan- 
t ii les, which is afterwards dh ided 
into pure and mixed mathemal Lcs. 

The branches of pure mat heniat ics 

are arithmetic, geometry, algebra, 
analytical geomt try, and the fflffi ren- 
Ual and Integral calculus; the three 
latter embrace I he entire portion of 
mathematical Bcience in which 
quantities are represented, not by 
numbers bul by letters ot the alpha- 
bet. 



J26 



Index and Useful Definitions. 



Mathematics, three departments of, 

337. 
Matter, 46. 
Maximum is the greatest number or 

quantity attainable in any given 

case; opposed to minimum. 
Mean proportional, 150. 
Means of a proportion, 150. 
Measures, seven kinds, 337. 
Measure, 13. 

Measurement of heat, 156. 
Mechanical powers, 48. 

Division of, 48. 
Mechanical philosophy, 45. 
Mechanical theory of heat, 153. 
Mechanism of steam engines, 353. 
Mechanics, 45. 
Melting points of alloys, 280. 
Melting point of solids, Table, 280. 
Mensuration, 93. 
M. E. P. Indicates Mean Effective 

Pressure. 
Mercury, specific gravity, 317. 
Mercury, melting point, 280. 
Metals, strength of, Table, 231. 
Metre, 39. 

Metric, or French system, Table, 39. 
Micro-farad, 290. 
Millimetre, 39. 
Minimum is the least quantity; 

opposed to maximum. 
Minuend, its arithmetical definition, 

12. 
Minutes, sign of, 32, 129. 
Miscellaneous measures, 39. 
Mixed Mathematics are the ap- 

applications of calculations to the 

objects of art and nature. The 

Hand Book of Calculations is an 

illustration of a work of mixed 

mathematics. 
Momentum, 254. 
Money, Unit of U. S., 37. 

How to write it, 37. 

Table of U. S., 37. 

Note, 91. 
Motion, 354. 
Multiplication, 35. 

Sign of, 12, 25. 

Table, 34. 

Rule for proof, 26. 

Examples, 35, 36. 

Same principle as addition, 35. 
Multiplicand, 13, 35. 
Multiplier, 13, 25. 
Nails, size of, 29L 

Number to the lb., 294. 
Natural philosophy, 45. 



Negative quantity, 222. 
Negative electricity, 289. 

N. H. P. Means Net Horse Power. 
Nitro-glycerine, melting point, 280. 
No. Abbreviation for Number. 
Non-condensing engines, 250. 
Nonagon, 133. 
Notation, 15. 

Examples, 23. 

Examples for practice, 16. 
Numbers, the unit of. 14. 
Number of days, Table, 295. 
Numeration, 15. 

Table, 15. 

Examples for practice, 17. 
Numerator, 106. 
%. Abbreviation for Per Cent. 
Oblique angles, 131. 
Oblique lines, 131. 
Oblong, 98. 

Rule to find the area of, 99. 

Example to illustrate rule, 99. 
Obtuse-angled triangle, 132. 
Obtuse angle, 131. 
Octagon, 132. 
Octahedron, 135. 
Ohm, The, 289, 290. 
Ohm meters, 291. 
Oils, specific gravity, 218. 
Open circuit, 387. 
Ordinate*, are the lines drawn across 

indicator diagrams to aid in calcu- 
lating their area. 
Oxygen, amount consumed, Table, 370. 
Packing ring, 356. 
Palm, (A,) 39. 
Paper measure, Table, 35. 
Parallel lines, 128. 
Parallelogram, 99, 133. 

Rule to find area, 99. 

Example to illustrate rule, 99. 
Parenthesis, 221. 
Pedestals, 264. 
Pentagon, 133. 
Perimeter, 134. 
Perimeter, in geometry, is the outer 

boundary of body or figure, or the 

sum of all the sides. In circular 

figures, instea d of perimeter we use 

circumference or periphery. 
Periphery, The circumference of a 

circle, ellipse, etc. 
Permutation, 91. 

Rule, 91. 

Examples, 91, 
Perpendicular, sign of, 129. 
Photometer, or light measurer, 270. 
Phosphorus, melting point, 280. 



Index and Useful Definitions. 



-: 



Pint, 30. 
Piston rod, ».">;. 

Maximum stress on, ~.">7. 
Piston*, steam, description, 856. 
I»it«h of tooth of wheels, 8L 
Piteh of propeller wheel, 80S. 

Method of ascertaining, 80S, 
PItoli oirolo or pitch line, 801. 
Piano figures, 131. 
Piano trigonometry) 138. 
Platinum, specific gravity, 817. 
Pneumatic*, 45. 
Point, (A. 
Polo pieces, 
Polycom, i:»l, 183. 
Positive electricity, 289. 
Positive quantity, 882. 
Postulate, (A,) 188. 

Example, 129. 
Potential, 887. 
Pound of steam, xW. 
Power, one candle, 869. 
Procure on stay bolts, Rule and 
example, 2>7. 

Tlie unit of, 14. 
Prime numbers are those divisable 

only by unity, or one. 
Primary powers, 47. 
Problem, 128.288. 
Product. Arithmetical definition, 12, 

£->. 
Proof of fctrenjrtli is that to which 

a boiler is Bnhjected to when being 

tested, and is usually considerably 
eater than tilt,' working load, so 

called: Example, A boiler designed 

to carry 125 Lbs. steam pressure is 

proved to I")" or more, cold water 

pressure. 

Propeller wheel, 301. 

Right and left hand, 808. 
Proportion, 149. 
Rule, ISO. 
Example, 160. 

•i of, It'.'. 

Note, 160. 
Proposition, 128. 
Pnllles, d loose, 64. 

Bole for speeds and Biases, 899. 

Belting, etc., 297. 
Pnlle] 

Movable, 

Fix. 
Exao | 
Pnmps. Notes, 188. 

Rule to And the capacity of a crater 
cylinder, 

Role to find the load on a pomp, 188, 



Pumps. Ride to find the Bteam 
pressure required, 185. 
Rule to find the diameter of cylinder 

required, 185. 
Rule to find the total amount of 

pressure, 186. 
Rule to find the resistance of water, 

1ST. 
Rule to find the H. P. required to 

raise water, 187. 
Rule to find the water capacity of a 

steam pump, 1S1. 
Rule to find the pressure in Lbs. per 
square inch of a column of water, 
188. 
Rule to find the height of a column 
of water in feet, the pressure being 
known, 183. 
Rule to find the horse power to 

pump water, 183. 
Rule to find the quantity of water 
pumped in 1 minute, 184. 
Pumps, ISO and 181. 
Friction of, 187. 
Slip of, 187. 
Leakage of, 187. 
Amount to be added, 187. 
Rules and examples, 181-190. 
Pyrometers, 157. 
Quadrilateral, 133. 
Quad rant, 32. 
Quantity, 13. 
Compound, .•J:.':.'. 

How expressed in algebra, 281. 
Positive, negative and simple, ZZl. 
Quotient, 12, 88. 
Had! us, 134. 

Radiation of heat, 151, h">2. 
Batlo, 1 19. 

Reaumur's thermometer, 157. 
Receiver, descrij tion, 850. 
Reciprocal. Acting alternately, or 

backwards and forwards, 'fids is 
one of the words used very frequent- 
ly in various parts of mathematics - 
hence Its primary meaning— back 

and forth motion-should be well 
tixed in tin; mind. 

Reciprocating motion, 882. 
Records tor engineers, 880. 
Reduction, 10. 

Ascending, 40. 
Descending, 40, 
Rule, i". 

Examples, I", n, 48, 18. 
Rule for prooving, 48. 

Note. II. 

Bis departments of. :rJ7. 



3^8 



Index ci7td Useful Definitions. 



Rectangle, 133. 

Regulation when used in reference to 
steam engines means that the ad- 
mission of steam to the cylinder is 
regulated, so as to insure uniformity 
of speed. 

Release. This term is understood to 
mean the exhaust. 

Resistance of electricity, 289. 

Rhombus, 133. 

Rhomboid, 133. 

Richard's Indicator, springs used 
in, 310. 

Right angle, 130. 

Right-angled triangle, 133. 

Roman notation, Table, 1 to 1 mill- 
ion, 38. 

Root of a power, 193. 

Rope, 76. 

Rules, 77, 80. 
Tables, 76, 77. 
Examples, 77, 80. 
Breaking strain, 80. 
Notes, 78, 80. 
Tensile strength of, 231. 

Rotary engines, 351. 

Rule of three, 149. 

Safe pressure for steam boiler, 336. 

Safety valve, 340. 
Calculating, 340-247. 
Calculations to test correctness of, 

347. 
Face of, 340. 

Safety valve spindle, description, 
240. 

Salino meter, 284. 

Salt water, boiling point, 283. 

Saturated sea water, 283. 

Saturated steam, 199. 

Scalene triangle, 131. 

Scholium, (A,) 128. 

Screw, 70. 
Rule, 71. 
Example, 71, 73. 

Seat of the safety valve, descrip- 
tion, 340. 

Secant of an arc, 137. 

Seconds, sig^i of, 33, 129. 

Setting slide valve, 305. 
Emergency rule, 306. 

Shearing stress, The, of iron is the 
strength which resists the action of 
cutting it across. Example, The 
rivets in a boiler are compelled to 
resist a shearing stress, as a boiler in 
exploding frequently cuts sharp 
across the rivets between the two 
boiler plates. 



Section of land, 36. 
Shaft bearings, 264. 

Signal code for engineers, 274. 
Silver, specific gravity, 217. 

Melting point, 280. 
Simple engines, 250. 
Simple numbers, definition, 43. 
Sine of an angle, 137. 

Abbreviation, 137. 
Size of nails, Table, 394. 
Slide bars of steam engine, 258. 
Slide valve, The, 303. 

Rule for setting, 304, 305. 

To scrieve lathe, 307. 
Slip of propeller wheel, 302. 
Solid, (A,) 46, 92, 128. 

Rectangular, Rule to find the con- 
tents, 103. 

Rectangular, Example, 103. 
Solids, surfaces of regular, Table, 135. 

Contents of, Table, 135. 
Space is boundless extension. 
Space, 160. 
Span, (A,) 39. 
Specific gravity, Law of, 215. 

Rules and example, 216, 218, 219, 330. 
Specific gravities. Table, 217. 
Specific gravity bottle, 215. 
Speed of toothed gears, rule for 

estimating, 301. 
Sphere, (A,) 102. 

Rule to find cubic contents, 105. 

Rule to find the surface, 102. 
Spindle of safety valve, descrip- 
tion, 240. 
Spring means the spring which is em- 
ployed in the cylinder of the indica- 
tor. 
Square root, 191. 

Rule and examples, 191, 192, 193. 

Table of, 275-279. 
Square, (A,) 133. 

Rule to find the area of, 98. 

Example to illustrate rule, 98. 
Squares, Table of, 275-279. 
Square foot. A space that is one 

foot wide, and one foot long. 
Statics treats of forces that keep bod- 
ies at rest or in equilibrium; opposed 

to dynamics. 
Steam room is that part of the boiler 

which is designed to contain the 

steam. This in the best practice is 

considered to be 1-4 to 1-6 of the 

whole. 
Steam expansion, 307, 307, 308. 

Gain from, 309. 

Table of volumes and pressures, 309. 



Index and Useful Definitions. 



329 



Statute mile, 38. 

Stay bolts, rule for strength, :?■>:. 

Strain, 196. 

Value of exhaust steam, 207. 

Saturated. 199. 

Total heat in steam. Rule and ex 
amples. 804, 90S. 

Latent heat of, 202. 

Speed of escaping, 805. 

Format ion of under pressure, 901,208. 

Superheated, 900. 

Properties of, 198. 

The moment or Leverage of, -44. 

Weight of, 20& 

Note, 201. 

To generate 1 Lb. at 212°, Table, 204. 

Properties of saturated. Table, 200. 
Steam boiler, 171,233. 

Rule for strength, 2;>o. 

Rule and example for water capaci- 
ty, 23a 

Heating surface. 171. 

Note relating to safety, 936. 
Steam cylinder, Rule for H. P., 251. 
Steam cylinder, rule to find capacity 

of, 253. 
Steam engine, description, 249. 
-ideation. 249. 

Connecting rods, 25S. 
• ■mors. 265, 266, 267. 
Steam engine counters, 268. 

Example of nse, 269. 
steam engine Indicator, 310. 
Steam fire engine, role for power, 170 
Steam pistons, 255. 
Steam space, Pule to compute, 2.59. 
Steel, melting point, 280. 

Te< Bile strength, 231. 

Specific gravity, ''17. 
Sterling or I English money, Table, 

Stones, specific gravity. 217. 
St rati n op journals, 263. 

on crank shaft, 
Strength of steam boiler**, Rule 

and example-, :.':>;. 
Strength of materials. 

Rule and examples, 252. 

Important principles, 889. 
Tablet 
Subtraction. 

Tabh 

Pul. 

Bole for proving corn 

Subtrahend. Arithmetical defini- 
tion, 12. 



Sulphur, melting point, 280. 
Sulphuric add, specific gravity, 218. 

Melting point, 280. 
Sum. Its arithmetical definition, 12. 
Supplement of an arc, 137. 
J Superheated steam, 200. 
Surface or square measure, Table, 

31. 
Surface, (A,) 92, 128. 

The unit of, 14. 
Symbols, 221. 
Tablespoonfnl, 39. 
Tallow, melting point, 280. 
Ta nirciit of an arc, 137. 

Abbreviation, 137. 

I 'cm »| 11 1 11 1, 39. 

Teeth of wheels, pitch of, 301. 
Teeth, hearing through, note, 307. 
Tensile strain or strength, 228. 
Terms of a ratio, 150. 
Terms of an algebraic expression, 222. 
Tetrahedron, a figure, 135. 
Theorem, (A,) 128. 

Abbreviation of, 129, 222. 
Therefore, arithmetical sign of, 222. 
Thermometers, 155. 
Theriiio-dynamies, 150. 
Tli rust , direct or compression, 228. 
Tide power, 47. 

Time is defined as endless succession. 
Time, 160. 

The unit of, 14. 
Time measure, Table, 35. 
Tin, melting point, 880. 

Specific gravity, 217. 
Torsion, or tensile strength, 229. 
Transmission of power, 296. 
Transverse. Lying or being across, 

turned across. 
Transverse strength or strain, 229. 
Trapezium, (A,) 100, 183. 

Rule to find the area, 100. 
Trapezoid, (A,) 101, 183. 

Pule to find the area, 101. 

Example to illustrate rule, 101. 
Triangle', 100, 131. 

Rule to find the area, loo. 

Sign Of, 129. 

Trigonometry treats of the measuro- 

iiicnt of t riangles. 
Trigonometry, 188. 
Troy freight, Table, -'A. 
Tumblerful, 39. 
Turpentine, melting point, 880. 
t it. Abbreviation for Last Mouth. 
Useful points relating to belting, 800. 
Utmost strength, The, of a boiler 

Ifl the point at which it will explode. 



330 



Index and Useful Definitions. 



"Unit, definition of, 13. 

Various kinds, 14. 

In notation and numeration, 15. 

Table of, 19. 

Of work, 154 

Of heat, 154. 

For measurement of light, 270. 

Of electrical measurements, 289. 
Useful numbers for engineers, 36. 
Value of figures, simple and local, 

227. 
Valve seat, to scrieve lathe, 307. 
Velocity, definition, 254. 
Velocity of steam, 205-206. 

Tables, 206. 

Note, 207. 
Vertical. Perpendicularly; over the 

head; being in a position, up and 

down, to the line of the horizo". 
Versed sine of an arc, 137. 
Vertex, 131. 
Vertical steam boiler, Rule and 

examples of, 234. 
Vinculum, 221. 
Vital action, power of, 47. 
Volcanic power, 47. 
Volt, definition, 290. 
Volt meters, 291. 
Vortex ring, is a ring having motion 

in a direct line, moving upward or 

otherwise, and revolving inwardly 

upon the axis of its circumference; 

a round rubber band about a stick, 

as the band is forced along the stick, 

will rotate inwardly and furnish an 

example of vortex motion. 
Vulgar fractions, 106, 107, 108. 
Wages, Table, 33. 

Examples, 43, 44. 
Water, 176. 

Standard of specific gravities, 215. 

Standard temperature, 176. 

Thermal units in 1 lb. of, Table, 208. 

Point of maximum density, 176. 

Diameters of pipes, Table, 190. 

Quantity discharged per minute, 
Table, 189. 

Freezing point, 176. 

Boiling point, 176. 

Weight of, 176, 177. 



Water, pressure of, at different heats, 
Table, 190. 
Table of pressures, 179. 
Increase of pressure, 179. 
Quantity discharged per minute by 

double cylinders, Table, 189. 
Important principles, 178, 179. 

Water capacity of the steam boiler, 
Rule and Example, 238. 

Water power, 47. 

Water space. This is the part of the 
steam boiler which contains the wa- 
ter, and is usually about ^ of the 
whole cubical contents. 

Watt, The, 291. 

Watt, James, inventor of the Indica- 
tor, 310. 

Wax, melting point, 280. 

Wedge, 73. 
Rule, 74, 75. 
Example, 74, 75. 

Wedgwood's pyrometer, 158. 

Weight, 212, 254. 

Weights of water, Table, 36. 

Weight of iron and steel shaft- 
ing, Table, 293. 

Weight of metals, Table, 36. 

Where to use capital letters, 317. 

Wheel and axle, 57. 

Rules and examples, 58, 59. 
Chinese wheel and axle, 59. 
Rule, 60. 
Examples, 60, 61. 

Wind power, 47. 

Wire, tensile strength of, 231. 

Wire rope, Tables, 79. 
Examples, 80. 
Notes, 78. 

Woods, specific gravity, 217, 218. 
Tensile strength, 231. 

Work, the unit of, 14. 

Working load, The, is the safe 
working pressure to which the boil- 
er is proportioned, and this is usual- 
ly estimated to be 1-6 of the bursting 
pressure or ultimate strength. 

Wrought iron, melting point, 280. 

Zero, 156. 

Zinc, melting point, 280. 
Specific gravity, 217. 



ERRATA. 

Please make notations of errors as follows in the first edi- 
tions: page 43, 167, 239 — none of which, happily, will he 
likely to mislead the stndent. 

On page 268, as the pointer for the last dial has been drawn, 
the instrument should be read 496,039 instead of 596,039. 

THE END . 



WHAT ENGINEERS SAY OF THE BOOK. 



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Mathematical Drawing Instalments. 



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Every draftsman finds a drawing board, a T-square, and one or 
more triangles, as essential to his work as does the new beginner in 
mechanical drawing. 

The New Handy Drawing Board '• outfit ,; is intended for use at 
home, in the class-room, office and shop its wide use will be productive 
of good to aspiring engineers and mechanics ; no investment can be 
more judicious than the small cost represented by this •' outfit," a 
few tools and a book of instruction on drawing. 

As shown in the illustration, the No. 1 outfit consists of a board 
about ten by twelve inchps, to which a pad of drawing-paper is fas- 
tened, and a wooden T-square and triangles of suitable size. The 



MATHEMATICAL DRAWING INSTRUMENTS. 



draftsman fastens the piece of paper on which he is working to his 
drawing-hoard by means of thumb-tacks, but this method 1ms proved 
both expensive and annoying, and so the scheme of the pad has been 
devised. This pad is slightly glued to the hoard at each, corner and 

the shorts composing it are torn off, one by one, as fast as they are 
used. 

The pads are sold separately from the boards and can be renewed 
as often as circumstances require, those which are commonly furnished 
being made of twenty sheets of a special light-tinted paper with a 
reasonably good " tooth.' which will take ink and bear the use of the 
rubber fairly well. 

The T-square is a substantial instrument, having a blade fourteen 
inches long. The head is adapted for use with the pad, as well as the 
single sheet, being unusually thick, so as to ^llow it to have a hold on 
the board when the pad is of full thickness. The two triangles, com- 
monly called the 45° and (50 triangles, include all the standard angles, 
00 , 45 . 00 and 30 , ordinarily needed by draftsmen. 

As a convenience in keeping the several pieces of the set together 
the back of the board is provided with grooved cleats and the cross 
cleats at the two ends of the board are slotted to receive the tongue of 
the T-square, so that when all the pieces are in place they are securely 
locked together, as shown. This arrangement makes it impossible 
tor any of the parts to be lost or broken while the board is not in use, 
provided they are properly packed in their places. 

Tin- No. 2 outfit is double the size of No. 1, being particularly de- 
d for advanced students and draftsmen. 

Tli.- element of aeeuracy has been very carefully considered in 
making the drawing outfit, and it is offere I at a price so low that no 
one can afford to do without it. 




Prices, post paid. 

Outfit N". l consists of Board (10xl2J inches), pad of 80 sheets 
drawing paper, i T Bquareand 3 triangles or set! squares 65cts. 



Outfit No. -i. consiste «»f Board (18x19 inches), pad of 30 sheets of 
drawing paper, l T square and 2 triangles oi Betl squares- $1.00. 



MATHEMATICAL DRAWING INSTRUMENTS. 




Triangular Scales. 

This scale, as shown in above illustration, is the most desirable 
article in its line that can possibly be imagined. The triangular, box- 
wood architect's scale has come to be a recognized necessity by every 
draftsman, but the cost has prevented its general use, and hence paper 
scales and other unsatisfactory substitutes have been devised. The 
scale is good enough for the regular use of the best architects and 
draftsmen. 

This entire line of rules and scales is made from the best rock-maple 
and the goods are carefully inspected and selected at each stage of 
manufacture and finally finished with a good hand-rubbed shellac 
polish. Each scale is provided with a manilla case, to serve as a per- 
manent protection for the edges. Price 50 cts. 



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13 3 4 


5 6 7 


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1 I i G l 1 III mill ilililililii^ ! filil 8 iiili!^MfeB# LLLlJ L,y '' 0J , 



Drawing Rule. 

This is a thin rule, one inch wide, with one edge beveled and gradu- 
ated to inches and sixteenths. In addition to its use for making 
measurements this wide, thin rule is designed to go with draftsmen's 
triangles in elementary industrial drawing. Price, 20 cts. 



Flat Drawing Scales. 

This set comprises three pieces, each of which has two beveled 
edges, and each edge contains a scale, so that the set gives six gradua- 
ated edges. 

No. 1 contains the 12-inch rule in sixteenths on one edge and the 
1-inch scale on the other edge. 

No. 2 contains the 3-inch scale and the li-inch scale. 

No. 3 contains the f-inch scale and the i-inch scale. 

Price, each, of Nos. 1, 2 or 3, 20 cts. 



CATALOGUE OF MECHANICAL DRAWING INSTRUMENTS. 



Set 1. — Fig. 40. Consists of excellent brass instruments: 1 divider 4^ 
inches: pen and pencil attachment : crayon holder; scale: Lengthening 
bar and protractor: in a polished mahogany case, and one ivory handled 
drawing pen not shown in cut. 

Price, $1.00. postpaid. 



Fig. 38. 



\ Ig. 35 




Set 8.— Fig. 45. Brass. 2 dividers 4i inches ; pen and pencil attach- 
ment : lengthening bar; rul- 
ing pen : crayon holder : scale 
and protractor; mahogany 
case; also extra fine 5J inch 
dividers and ivory handled 
drawing pen (figs. 32 and 35). 
Price, $2.00, free of post- 
age. 

Set 3.— Fig. 50. German 

Silver in round cornered case, 
cloth, suitable for the pocket. 
Contains 6 inch divider : rul- 

Fig. 40. 

ing pen ; pen and pencil at- 
tachment ; protractor and 
rules, and set square. 

Price, $3.00, postpaid. 

Extra Fine Drawing 
Pen. — Fig. 32. This has an 
ivory handle, and is designed 
for a working-tool for life 
service. Price, 25 Gents. 
This tool is included in sets 1 
and 2, but is sold separate. 







Extra Fine Dividers. — 
B5. These are <• h o i c e 
working instruments for long 
lervice, and are added to the 
tools in set -'. and also sold 
separate. 
Price, 25 Cents, postpaid. 



No working eng ineer 
■hould be without a few draw- 
re of- 
fered as hein^ t he hesi Cor the 

■ vet Bold. 




Big. 60. 



336 



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CATALOGUE OF MECHANICAL DRAWING INSTRUMENTS. 



The following sets are extremely beautiful in style and finish, the boxes 
being of fine leather and the instruments of German silver. 




Fig. 50. 

Set 4.— Morocco Box: containing pair of 5|-inch Dividers, with Pen, 
Pencil, and Needle Points and Lengthening Bar. 
Pair of 5-inch plain Dividers. 

Pair of 4- inch Dividers, with Pen, Pencil and Needle Points. 
2 Drawing Pens. . .v price $6.50 




Set 5.— Morocco Box; cantaining pair of 5|-mch Dividers, with Pen, 
Pencil, and Needle Points and Lengthening Bar. 
Pair of 5-inch plain Dividers. . 

Pair of 4-inch Dividers, with Pen, Pencil, and Needle Points. 
Pair Spacing Dividers. 
Bow Pen. 

Bow Pencil. 41ft A A 

2 Drawing Pens price $10.00 



These goods are sent (post free) to all parts of the United States 

and Canada on receipt of price. Postage stamps received jor 

small sums. Money or express orders, or checks payable to oraer 

preferred. Order of us direct, thus avoiding delay. 

THEO. AXJI>E1L. «&> CO., Fulfrlissliers, <&e. 9 

91 Liberty Street, Office 3, 

NEW YORK. 



021 178 002 2 






